Theory AOT_PLM

1(*<*)
2theory AOT_PLM
3  imports AOT_axioms
4begin
5(*>*)
6
7section‹The Deductive System PLM›
8
9(* constrain sledgehammer to the abstraction layer *)
10unbundle AOT_no_atp
11
12(* To enable meta syntax: *)
13(*interpretation AOT_meta_syntax.*)
14(* To disable meta syntax: *)
15interpretation AOT_no_meta_syntax.
16
17(* To enable AOT syntax (takes precedence over meta syntax; can be done locally using "including" or "include"): *)
18unbundle AOT_syntax
19(* To disable AOT syntax (restoring meta syntax or no syntax; can be done locally using "including" or "include"): *)
20(* unbundle AOT_no_syntax *)
21
22AOT_theorem "modus-ponens": assumes ‹φ› and ‹φ → ψ› shows ‹ψ›
23  using assms by (simp add: AOT_sem_imp) (* NOTE: semantics needed *)
24lemmas MP = "modus-ponens"
25
26AOT_theorem "non-con-thm-thm": assumes ‹⊢□ φ› shows ‹⊢ φ›
27  using assms by simp
28
29AOT_theorem "vdash-properties:1[1]": assumes ‹φ ∈ Λ› shows ‹⊢ φ›
30  using assms unfolding AOT_model_act_axiom_def by blast (* NOTE: semantics needed *)
31
32attribute_setup act_axiom_inst =
33  ‹Scan.succeed (Thm.rule_attribute [] (K (fn thm => thm RS @{thm "vdash-properties:1[1]"})))›
34  "Instantiate modally fragile axiom as modally fragile theorem."
35
36AOT_theorem "vdash-properties:1[2]": assumes ‹φ ∈ Λ□› shows ‹⊢□ φ›
37  using assms unfolding AOT_model_axiom_def by blast (* NOTE: semantics needed *)
38
39attribute_setup axiom_inst =
40  ‹Scan.succeed (Thm.rule_attribute [] (K (fn thm => thm RS @{thm "vdash-properties:1[2]"})))›
41  "Instantiate axiom as theorem."
42
43method cqt_2_lambda_inst_prover = (fast intro: AOT_instance_of_cqt_2_intro)
44method "cqt:2[lambda]" = (rule "cqt:2[lambda]"[axiom_inst]; cqt_2_lambda_inst_prover)
45lemmas "cqt:2" = "cqt:2[const_var]"[axiom_inst] "cqt:2[lambda]"[axiom_inst] AOT_instance_of_cqt_2_intro
46method "cqt:2" = (safe intro!: "cqt:2")
47
48AOT_theorem "vdash-properties:3": assumes ‹⊢□ φ› shows ‹Γ ⊢ φ›
49  using assms by blast
50
51AOT_theorem "vdash-properties:5": assumes ‹Γ1 ⊢ φ› and ‹Γ2 ⊢ φ → ψ› shows ‹Γ1, Γ2 ⊢ ψ›
52  using MP assms by blast
53
54AOT_theorem "vdash-properties:6": assumes ‹φ› and ‹φ → ψ› shows ‹ψ›
55  using MP assms by blast
56
57AOT_theorem "vdash-properties:8": assumes ‹Γ ⊢ φ› and ‹φ ⊢ ψ› shows ‹Γ ⊢ ψ›
58  using assms by argo
59
60AOT_theorem "vdash-properties:9": assumes ‹φ› shows ‹ψ → φ›
61  using MP "pl:1"[axiom_inst] assms by blast
62
63AOT_theorem "vdash-properties:10": assumes ‹φ → ψ› and ‹φ› shows ‹ψ›
64  using MP assms by blast
65lemmas "→E" = "vdash-properties:10"
66
67AOT_theorem "rule-gen": assumes ‹for arbitrary α: φ{α}› shows ‹∀α φ{α}›
68  using assms by (metis AOT_var_of_term_inverse AOT_sem_denotes AOT_sem_forall) (* NOTE: semantics needed *)
69lemmas GEN = "rule-gen"
70
71AOT_theorem "RN[prem]": assumes ‹Γ ⊢□ φ› shows ‹□Γ ⊢□ □φ›
72  by (meson AOT_sem_box assms image_iff) (* NOTE: semantics needed *)
73AOT_theorem RN: assumes ‹⊢□ φ› shows ‹□φ›
74  using "RN[prem]" assms by blast
75
76AOT_axiom "df-rules-formulas[1]": assumes ‹φ ≡df ψ› shows ‹φ → ψ›
77  using assms by (simp_all add: AOT_model_axiomI AOT_model_equiv_def AOT_sem_imp) (* NOTE: semantics needed *)
78AOT_axiom "df-rules-formulas[2]": assumes ‹φ ≡df ψ› shows ‹ψ → φ›
79  using assms by (simp_all add: AOT_model_axiomI AOT_model_equiv_def AOT_sem_imp) (* NOTE: semantics needed *)
80(* NOTE: for convenience also state the above as regular theorems *)
81AOT_theorem "df-rules-formulas[3]": assumes ‹φ ≡df ψ› shows ‹φ → ψ›
82  using "df-rules-formulas[1]"[axiom_inst, OF assms].
83AOT_theorem "df-rules-formulas[4]": assumes ‹φ ≡df ψ› shows ‹ψ → φ›
84  using "df-rules-formulas[2]"[axiom_inst, OF assms].
85
86
87AOT_axiom "df-rules-terms[1]":
88  assumes ‹τ{α1...αn} =df σ{α1...αn}›
89  shows ‹(σ{τ1...τn}↓ → τ{τ1...τn} = σ{τ1...τn}) & (¬σ{τ1...τn}↓ → ¬τ{τ1...τn}↓)›
90  using assms by (simp add: AOT_model_axiomI AOT_sem_conj AOT_sem_imp AOT_sem_eq AOT_sem_not AOT_sem_denotes AOT_model_id_def) (* NOTE: semantics needed *)
91AOT_axiom "df-rules-terms[2]":
92  assumes ‹τ =df σ›
93  shows ‹(σ↓ → τ = σ) & (¬σ↓ → ¬τ↓)›
94  by (metis "df-rules-terms[1]" case_unit_Unity assms)
95(* NOTE: for convenience also state the above as regular theorems *)
96AOT_theorem "df-rules-terms[3]":
97  assumes ‹τ{α1...αn} =df σ{α1...αn}›
98  shows ‹(σ{τ1...τn}↓ → τ{τ1...τn} = σ{τ1...τn}) & (¬σ{τ1...τn}↓ → ¬τ{τ1...τn}↓)›
99  using "df-rules-terms[1]"[axiom_inst, OF assms].
100AOT_theorem "df-rules-terms[4]":
101  assumes ‹τ =df σ›
102  shows ‹(σ↓ → τ = σ) & (¬σ↓ → ¬τ↓)›
103  using "df-rules-terms[2]"[axiom_inst, OF assms].
104
105
106AOT_theorem "if-p-then-p": ‹φ → φ›
107  by (meson "pl:1"[axiom_inst] "pl:2"[axiom_inst] MP)
108
109AOT_theorem "deduction-theorem": assumes ‹φ ⊢ ψ› shows ‹φ → ψ›
110  using assms by (simp add: AOT_sem_imp) (* NOTE: semantics needed *)
111lemmas CP = "deduction-theorem"
112lemmas "→I" = "deduction-theorem"
113
114AOT_theorem "ded-thm-cor:1": assumes ‹Γ1 ⊢ φ → ψ› and ‹Γ2 ⊢ ψ → χ› shows ‹Γ1, Γ2 ⊢ φ → χ›
115  using "→E" "→I" assms by blast
116AOT_theorem "ded-thm-cor:2": assumes ‹Γ1 ⊢ φ → (ψ → χ)› and ‹Γ2 ⊢ ψ› shows ‹Γ1, Γ2 ⊢ φ → χ›
117  using "→E" "→I" assms by blast
118
119AOT_theorem "ded-thm-cor:3": assumes ‹φ → ψ› and ‹ψ → χ› shows ‹φ → χ›
120  using "→E" "→I" assms by blast
121declare "ded-thm-cor:3"[trans]
122AOT_theorem "ded-thm-cor:4": assumes ‹φ → (ψ → χ)› and ‹ψ› shows ‹φ → χ›
123  using "→E" "→I" assms by blast
124
125lemmas "Hypothetical Syllogism" = "ded-thm-cor:3"
126
127AOT_theorem "useful-tautologies:1": ‹¬¬φ → φ›
128  by (metis "pl:3"[axiom_inst] "→I" "Hypothetical Syllogism")
129AOT_theorem "useful-tautologies:2": ‹φ → ¬¬φ›
130  by (metis "pl:3"[axiom_inst] "→I" "ded-thm-cor:4")
131AOT_theorem "useful-tautologies:3": ‹¬φ → (φ → ψ)›
132  by (meson "ded-thm-cor:4" "pl:3"[axiom_inst] "→I")
133AOT_theorem "useful-tautologies:4": ‹(¬ψ → ¬φ) → (φ → ψ)›
134  by (meson "pl:3"[axiom_inst] "Hypothetical Syllogism" "→I")
135AOT_theorem "useful-tautologies:5": ‹(φ → ψ) → (¬ψ → ¬φ)›
136  by (metis "useful-tautologies:4" "Hypothetical Syllogism" "→I")
137
138AOT_theorem "useful-tautologies:6": ‹(φ → ¬ψ) → (ψ → ¬φ)›
139  by (metis "→I" MP "useful-tautologies:4")
140
141AOT_theorem "useful-tautologies:7": ‹(¬φ → ψ) → (¬ψ → φ)›
142  by (metis "→I" MP "useful-tautologies:3" "useful-tautologies:5")
143
144AOT_theorem "useful-tautologies:8": ‹φ → (¬ψ → ¬(φ → ψ))›
145  by (metis "→I" MP "useful-tautologies:5")
146
147AOT_theorem "useful-tautologies:9": ‹(φ → ψ) → ((¬φ → ψ) → ψ)›
148  by (metis "→I" MP "useful-tautologies:6")
149
150AOT_theorem "useful-tautologies:10": ‹(φ → ¬ψ) → ((φ → ψ) → ¬φ)›
151  by (metis "→I" MP "pl:3"[axiom_inst])
152
153AOT_theorem "dn-i-e:1": assumes ‹φ› shows ‹¬¬φ›
154  using MP "useful-tautologies:2" assms by blast
155lemmas "¬¬I" = "dn-i-e:1"
156AOT_theorem "dn-i-e:2": assumes ‹¬¬φ› shows ‹φ›
157  using MP "useful-tautologies:1" assms by blast
158lemmas "¬¬E" = "dn-i-e:2"
159
160AOT_theorem "modus-tollens:1": assumes ‹φ → ψ› and ‹¬ψ› shows ‹¬φ›
161  using MP "useful-tautologies:5" assms by blast
162AOT_theorem "modus-tollens:2": assumes ‹φ → ¬ψ› and ‹ψ› shows ‹¬φ›
163  using "¬¬I" "modus-tollens:1" assms by blast
164lemmas MT = "modus-tollens:1" "modus-tollens:2"
165
166AOT_theorem "contraposition:1[1]": assumes ‹φ → ψ› shows ‹¬ψ → ¬φ›
167  using "→I" MT(1) assms by blast
168AOT_theorem "contraposition:1[2]": assumes ‹¬ψ → ¬φ› shows ‹φ → ψ›
169  using "→I" "¬¬E" MT(2) assms by blast
170
171AOT_theorem "contraposition:2": assumes ‹φ → ¬ψ› shows ‹ψ → ¬φ›
172  using "→I" MT(2) assms by blast
173
174(* TODO: this is actually a mixture of the two variants given in PLM; adjust. *)
175AOT_theorem "reductio-aa:1":
176  assumes ‹¬φ ⊢ ¬ψ› and ‹¬φ ⊢ ψ› shows ‹φ›
177  using "→I" "¬¬E" MT(2) assms by blast
178AOT_theorem "reductio-aa:2":
179  assumes ‹φ ⊢ ¬ψ› and ‹φ ⊢ ψ› shows ‹¬φ›
180  using "reductio-aa:1" assms by blast
181lemmas "RAA" = "reductio-aa:1" "reductio-aa:2"
182
183AOT_theorem "exc-mid": ‹φ ∨ ¬φ›
184  using "df-rules-formulas[4]" "if-p-then-p" MP "conventions:2" by blast
185
186AOT_theorem "non-contradiction": ‹¬(φ & ¬φ)›
187  using "df-rules-formulas[3]" MT(2) "useful-tautologies:2" "conventions:1" by blast
188
189AOT_theorem "con-dis-taut:1": ‹(φ & ψ) → φ›
190  by (meson "→I" "df-rules-formulas[3]" MP RAA(1) "conventions:1")
191AOT_theorem "con-dis-taut:2": ‹(φ & ψ) → ψ›
192  by (metis "→I" "df-rules-formulas[3]" MT(2) RAA(2) "¬¬E" "conventions:1")
193lemmas "Conjunction Simplification" = "con-dis-taut:1" "con-dis-taut:2"
194
195AOT_theorem "con-dis-taut:3": ‹φ → (φ ∨ ψ)›
196  by (meson "contraposition:1[2]" "df-rules-formulas[4]" MP "→I" "conventions:2")
197AOT_theorem "con-dis-taut:4": ‹ψ → (φ ∨ ψ)›
198  using "Hypothetical Syllogism" "df-rules-formulas[4]" "pl:1"[axiom_inst] "conventions:2" by blast
199lemmas "Disjunction Addition" = "con-dis-taut:3" "con-dis-taut:4"
200
201AOT_theorem "con-dis-taut:5": ‹φ → (ψ → (φ & ψ))›
202  by (metis "contraposition:2" "Hypothetical Syllogism" "→I" "df-rules-formulas[4]" "conventions:1")
203lemmas Adjunction = "con-dis-taut:5"
204
205AOT_theorem "con-dis-taut:6": ‹(φ & φ) ≡ φ›
206  by (metis Adjunction "→I" "df-rules-formulas[4]" MP "Conjunction Simplification"(1) "conventions:3")
207lemmas "Idempotence of &" = "con-dis-taut:6"
208
209AOT_theorem "con-dis-taut:7": ‹(φ ∨ φ) ≡ φ›
210proof -
211  {
212    AOT_assume ‹φ ∨ φ›
213    AOT_hence ‹¬φ → φ›
214      using "conventions:2"[THEN "df-rules-formulas[3]"] MP by blast
215    AOT_hence ‹φ› using "if-p-then-p" RAA(1) MP by blast
216  }
217  moreover {
218    AOT_assume ‹φ›
219    AOT_hence ‹φ ∨ φ› using "Disjunction Addition"(1) MP by blast
220  }
221  ultimately AOT_show ‹(φ ∨ φ) ≡ φ›
222    using "conventions:3"[THEN "df-rules-formulas[4]"] MP
223    by (metis Adjunction "→I")
224qed
225lemmas "Idempotence of ∨" = "con-dis-taut:7"
226
227
228AOT_theorem "con-dis-i-e:1": assumes ‹φ› and ‹ψ› shows ‹φ & ψ›
229  using Adjunction MP assms by blast
230lemmas "&I" = "con-dis-i-e:1"
231
232AOT_theorem "con-dis-i-e:2:a": assumes ‹φ & ψ› shows ‹φ›
233  using "Conjunction Simplification"(1) MP assms by blast
234AOT_theorem "con-dis-i-e:2:b": assumes ‹φ & ψ› shows ‹ψ›
235  using "Conjunction Simplification"(2) MP assms by blast
236lemmas "&E" = "con-dis-i-e:2:a" "con-dis-i-e:2:b"
237
238AOT_theorem "con-dis-i-e:3:a": assumes ‹φ› shows ‹φ ∨ ψ›
239  using "Disjunction Addition"(1) MP assms by blast
240AOT_theorem "con-dis-i-e:3:b": assumes ‹ψ› shows ‹φ ∨ ψ›
241  using "Disjunction Addition"(2) MP assms by blast
242AOT_theorem "con-dis-i-e:3:c": assumes ‹φ ∨ ψ› and ‹φ → χ› and ‹ψ → Θ› shows ‹χ ∨ Θ›
243  by (metis "con-dis-i-e:3:a" "Disjunction Addition"(2) "df-rules-formulas[3]" MT(1) RAA(1) "conventions:2" assms)
244lemmas "∨I" = "con-dis-i-e:3:a" "con-dis-i-e:3:b" "con-dis-i-e:3:c"
245
246AOT_theorem "con-dis-i-e:4:a": assumes ‹φ ∨ ψ› and ‹φ → χ› and ‹ψ → χ› shows ‹χ›
247  by (metis MP RAA(2) "df-rules-formulas[3]" "conventions:2" assms)
248AOT_theorem "con-dis-i-e:4:b": assumes ‹φ ∨ ψ› and ‹¬φ› shows ‹ψ›
249  using "con-dis-i-e:4:a" RAA(1) "→I" assms by blast
250AOT_theorem "con-dis-i-e:4:c": assumes ‹φ ∨ ψ› and ‹¬ψ› shows ‹φ›
251  using "con-dis-i-e:4:a" RAA(1) "→I" assms by blast
252lemmas "∨E" = "con-dis-i-e:4:a" "con-dis-i-e:4:b" "con-dis-i-e:4:c"
253
254AOT_theorem "raa-cor:1": assumes ‹¬φ ⊢ ψ & ¬ψ› shows ‹φ›
255  using "&E" "∨E"(3) "∨I"(2) RAA(2) assms by blast
256AOT_theorem "raa-cor:2": assumes ‹φ ⊢ ψ & ¬ψ› shows ‹¬φ›
257  using "raa-cor:1" assms by blast
258AOT_theorem "raa-cor:3": assumes ‹φ› and ‹¬ψ ⊢ ¬φ› shows ‹ψ›
259  using RAA assms by blast
260AOT_theorem "raa-cor:4": assumes ‹¬φ› and ‹¬ψ ⊢ φ› shows ‹ψ›
261  using RAA assms by blast
262AOT_theorem "raa-cor:5": assumes ‹φ› and ‹ψ ⊢ ¬φ› shows ‹¬ψ›
263  using RAA assms by blast
264AOT_theorem "raa-cor:6": assumes ‹¬φ› and ‹ψ ⊢ φ› shows ‹¬ψ›
265  using RAA assms by blast
266
267(* TODO: note these need manual introduction rules *)
268AOT_theorem "oth-class-taut:1:a": ‹(φ → ψ) ≡ ¬(φ & ¬ψ)›
269  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
270     (metis "&E" "&I" "raa-cor:3" "→I" MP)
271AOT_theorem "oth-class-taut:1:b": ‹¬(φ → ψ) ≡ (φ & ¬ψ)›
272  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
273     (metis "&E" "&I" "raa-cor:3" "→I" MP)
274AOT_theorem "oth-class-taut:1:c": ‹(φ → ψ) ≡ (¬φ ∨ ψ)›
275  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
276     (metis "&I" "∨I"(1, 2) "∨E"(3) "→I" MP "raa-cor:1")
277
278AOT_theorem "oth-class-taut:2:a": ‹(φ & ψ) ≡ (ψ & φ)›
279  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
280     (meson "&I" "&E" "→I")
281lemmas "Commutativity of &" = "oth-class-taut:2:a"
282AOT_theorem "oth-class-taut:2:b": ‹(φ & (ψ & χ)) ≡ ((φ & ψ) & χ)›
283  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
284     (metis "&I" "&E" "→I")
285lemmas "Associativity of &" = "oth-class-taut:2:b"
286AOT_theorem "oth-class-taut:2:c": ‹(φ ∨ ψ) ≡ (ψ ∨ φ)›
287  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
288     (metis "&I" "∨I"(1, 2) "∨E"(1) "→I")
289lemmas "Commutativity of ∨" = "oth-class-taut:2:c"
290AOT_theorem "oth-class-taut:2:d": ‹(φ ∨ (ψ ∨ χ)) ≡ ((φ ∨ ψ) ∨ χ)›
291  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
292     (metis "&I" "∨I"(1, 2) "∨E"(1) "→I")
293lemmas "Associativity of ∨" = "oth-class-taut:2:d"
294AOT_theorem "oth-class-taut:2:e": ‹(φ ≡ ψ) ≡ (ψ ≡ φ)›
295  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"]; rule "&I";
296      metis "&I" "df-rules-formulas[4]" "conventions:3" "&E" "Hypothetical Syllogism" "→I" "df-rules-formulas[3]")
297lemmas "Commutativity of ≡" = "oth-class-taut:2:e"
298AOT_theorem "oth-class-taut:2:f": ‹(φ ≡ (ψ ≡ χ)) ≡ ((φ ≡ ψ) ≡ χ)›
299  using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
300        "→I" "→E" "&E" "&I"
301  by metis
302lemmas "Associativity of ≡" = "oth-class-taut:2:f"
303
304AOT_theorem "oth-class-taut:3:a": ‹φ ≡ φ›
305  using "&I" "vdash-properties:6" "if-p-then-p" "df-rules-formulas[4]" "conventions:3" by blast
306AOT_theorem "oth-class-taut:3:b": ‹φ ≡ ¬¬φ›
307  using "&I" "useful-tautologies:1" "useful-tautologies:2" "vdash-properties:6" "df-rules-formulas[4]" "conventions:3" by blast
308AOT_theorem "oth-class-taut:3:c": ‹¬(φ ≡ ¬φ)›
309  by (metis "&E" "→E" RAA "df-rules-formulas[3]" "conventions:3")
310
311AOT_theorem "oth-class-taut:4:a": ‹(φ → ψ) → ((ψ → χ) → (φ → χ))›
312  by (metis "→E" "→I")
313AOT_theorem "oth-class-taut:4:b": ‹(φ ≡ ψ) ≡ (¬φ ≡ ¬ψ)›
314  using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
315        "→I" "→E" "&E" "&I" RAA by metis
316AOT_theorem "oth-class-taut:4:c": ‹(φ ≡ ψ) → ((φ → χ) ≡ (ψ → χ))›
317  using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
318        "→I" "→E" "&E" "&I" by metis
319AOT_theorem "oth-class-taut:4:d": ‹(φ ≡ ψ) → ((χ → φ) ≡ (χ → ψ))›
320  using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
321        "→I" "→E" "&E" "&I" by metis
322AOT_theorem "oth-class-taut:4:e": ‹(φ ≡ ψ) → ((φ & χ) ≡ (ψ & χ))›
323  using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
324        "→I" "→E" "&E" "&I" by metis
325AOT_theorem "oth-class-taut:4:f": ‹(φ ≡ ψ) → ((χ & φ) ≡ (χ & ψ))›
326  using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
327        "→I" "→E" "&E" "&I" by metis
328(* TODO: nicer proof *)
329AOT_theorem "oth-class-taut:4:g": ‹(φ ≡ ψ) ≡ ((φ & ψ) ∨ (¬φ & ¬ψ))›
330  apply (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"]; rule "&I"; rule "→I")
331   apply (drule "conventions:3"[THEN "df-rules-formulas[3]", THEN "→E"])
332   apply (metis "&I" "&E" "∨I"(1,2) MT(1) "raa-cor:3")
333  apply (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"]; rule "&I"; rule "→I")
334  using "&E" "∨E"(2) "raa-cor:3" by blast+
335AOT_theorem "oth-class-taut:4:h": ‹¬(φ ≡ ψ) ≡ ((φ & ¬ψ) ∨ (¬φ & ψ))›
336  apply (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"]; rule "&I"; rule "→I")
337  apply (metis "&I" "∨I"(1, 2) "→I" MT(1) "df-rules-formulas[4]" "raa-cor:3" "conventions:3")
338  by (metis "&E" "∨E"(2) "→E" "df-rules-formulas[3]" "raa-cor:3" "conventions:3")
339AOT_theorem "oth-class-taut:5:a": ‹(φ & ψ) ≡ ¬(¬φ ∨ ¬ψ)›
340  using "conventions:3"[THEN "df-rules-formulas[4]"]
341        "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
342AOT_theorem "oth-class-taut:5:b": ‹(φ ∨ ψ) ≡ ¬(¬φ & ¬ψ)›
343  using "conventions:3"[THEN "df-rules-formulas[4]"]
344        "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
345AOT_theorem "oth-class-taut:5:c": ‹¬(φ & ψ) ≡ (¬φ ∨ ¬ψ)›
346  using "conventions:3"[THEN "df-rules-formulas[4]"]
347        "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
348AOT_theorem "oth-class-taut:5:d": ‹¬(φ ∨ ψ) ≡ (¬φ & ¬ψ)›
349  using "conventions:3"[THEN "df-rules-formulas[4]"]
350        "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
351
352lemmas DeMorgan = "oth-class-taut:5:c" "oth-class-taut:5:d"
353
354AOT_theorem "oth-class-taut:6:a": ‹(φ & (ψ ∨ χ)) ≡ ((φ & ψ) ∨ (φ & χ))›
355  using "conventions:3"[THEN "df-rules-formulas[4]"]
356        "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
357AOT_theorem "oth-class-taut:6:b": ‹(φ ∨ (ψ & χ)) ≡ ((φ ∨ ψ) & (φ ∨ χ))›
358  using "conventions:3"[THEN "df-rules-formulas[4]"]
359        "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
360
361AOT_theorem "oth-class-taut:7:a": ‹((φ & ψ) → χ) → (φ → (ψ → χ))›
362  by (metis "&I" "→E" "→I")
363lemmas Exportation = "oth-class-taut:7:a"
364AOT_theorem "oth-class-taut:7:b": ‹(φ → (ψ →χ)) → ((φ & ψ) → χ)›
365  by (metis "&E" "→E" "→I")
366lemmas Importation = "oth-class-taut:7:b"
367
368AOT_theorem "oth-class-taut:8:a": ‹(φ → (ψ → χ)) ≡ (ψ → (φ → χ))›
369  using "conventions:3"[THEN "df-rules-formulas[4]"] "→I" "→E" "&E" "&I" by metis
370lemmas Permutation = "oth-class-taut:8:a"
371AOT_theorem "oth-class-taut:8:b": ‹(φ → ψ) → ((φ → χ) → (φ → (ψ & χ)))›
372  by (metis "&I" "→E" "→I")
373lemmas Composition = "oth-class-taut:8:b"
374AOT_theorem "oth-class-taut:8:c": ‹(φ → χ) → ((ψ → χ) → ((φ ∨ ψ) → χ))›
375  by (metis "∨E"(2) "→E" "→I" RAA(1))
376AOT_theorem "oth-class-taut:8:d": ‹((φ → ψ) & (χ → Θ)) → ((φ & χ) → (ψ & Θ))›
377  by (metis "&E" "&I" "→E" "→I")
378lemmas "Double Composition" = "oth-class-taut:8:d"
379AOT_theorem "oth-class-taut:8:e": ‹((φ & ψ) ≡ (φ & χ)) ≡ (φ → (ψ ≡ χ))›
380  by (metis "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
381            "→I" "→E" "&E" "&I")
382AOT_theorem "oth-class-taut:8:f": ‹((φ & ψ) ≡ (χ & ψ)) ≡ (ψ → (φ ≡ χ))›
383  by (metis "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
384            "→I" "→E" "&E" "&I")
385AOT_theorem "oth-class-taut:8:g": ‹(ψ ≡ χ) → ((φ ∨ ψ) ≡ (φ ∨ χ))›
386  by (metis "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
387            "→I" "→E" "&E" "&I" "∨I" "∨E"(1))
388AOT_theorem "oth-class-taut:8:h": ‹(ψ ≡ χ) → ((ψ ∨ φ) ≡ (χ ∨ φ))›
389  by (metis "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
390            "→I" "→E" "&E" "&I" "∨I" "∨E"(1))
391AOT_theorem "oth-class-taut:8:i": ‹(φ ≡ (ψ & χ)) → (ψ → (φ ≡ χ))›
392  by (metis "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
393            "→I" "→E" "&E" "&I")
394
395AOT_theorem "intro-elim:1": assumes ‹φ ∨ ψ› and ‹φ ≡ χ› and ‹ψ ≡ Θ› shows ‹χ ∨ Θ›
396  by (metis assms "∨I"(1, 2) "∨E"(1) "conventions:3"[THEN "df-rules-formulas[3]"] "→I" "→E" "&E"(1))
397
398AOT_theorem "intro-elim:2": assumes ‹φ → ψ› and ‹ψ → φ› shows ‹φ ≡ ψ›
399  by (meson "&I" "conventions:3" "df-rules-formulas[4]" MP assms)
400lemmas "≡I" = "intro-elim:2"
401
402AOT_theorem "intro-elim:3:a": assumes ‹φ ≡ ψ› and ‹φ› shows ‹ψ›
403  by (metis "∨I"(1) "→I" "∨E"(1) "intro-elim:1" assms)
404AOT_theorem "intro-elim:3:b": assumes ‹φ ≡ ψ› and ‹ψ› shows ‹φ›
405  using "intro-elim:3:a" "Commutativity of ≡" assms by blast
406AOT_theorem "intro-elim:3:c": assumes ‹φ ≡ ψ› and ‹¬φ› shows ‹¬ψ›
407  using "intro-elim:3:b" "raa-cor:3" assms by blast
408AOT_theorem "intro-elim:3:d": assumes ‹φ ≡ ψ› and ‹¬ψ› shows ‹¬φ›
409  using "intro-elim:3:a" "raa-cor:3" assms by blast
410AOT_theorem "intro-elim:3:e": assumes ‹φ ≡ ψ› and ‹ψ ≡ χ› shows ‹φ ≡ χ›
411  by (metis "≡I" "→I" "intro-elim:3:a" "intro-elim:3:b" assms)
412declare "intro-elim:3:e"[trans]
413AOT_theorem "intro-elim:3:f": assumes ‹φ ≡ ψ› and ‹φ ≡ χ› shows ‹χ ≡ ψ›
414  by (metis "≡I" "→I" "intro-elim:3:a" "intro-elim:3:b" assms)
415lemmas "≡E" = "intro-elim:3:a" "intro-elim:3:b" "intro-elim:3:c" "intro-elim:3:d" "intro-elim:3:e" "intro-elim:3:f"
416
417declare "Commutativity of ≡"[THEN "≡E"(1), sym]
418
419AOT_theorem "rule-eq-df:1": assumes ‹φ ≡df ψ› shows ‹φ ≡ ψ›
420  by (simp add: "≡I" "df-rules-formulas[3]" "df-rules-formulas[4]" assms)
421lemmas "≡Df" = "rule-eq-df:1"
422AOT_theorem "rule-eq-df:2": assumes ‹φ ≡df ψ› and ‹φ› shows ‹ψ›
423  using "≡Df" "≡E"(1) assms by blast
424lemmas "≡dfE" = "rule-eq-df:2"
425AOT_theorem "rule-eq-df:3": assumes ‹φ ≡df ψ› and ‹ψ› shows ‹φ›
426  using "≡Df" "≡E"(2) assms by blast
427lemmas "≡dfI" = "rule-eq-df:3"
428
429AOT_theorem  "df-simplify:1": assumes ‹φ ≡ (ψ & χ)› and ‹ψ› shows ‹φ ≡ χ›
430  by (metis "&E"(2) "&I" "≡E"(1, 2) "≡I" "→I" assms)
431(* TODO: this is a slight variation from PLM *)
432AOT_theorem  "df-simplify:2": assumes ‹φ ≡ (ψ & χ)› and ‹χ› shows ‹φ ≡ ψ›
433  by (metis "&E"(1) "&I" "≡E"(1, 2) "≡I" "→I" assms)
434lemmas "≡S" = "df-simplify:1"  "df-simplify:2"
435
436AOT_theorem "rule-ui:1": assumes ‹∀α φ{α}› and ‹τ↓› shows ‹φ{τ}›
437  using "→E" "cqt:1"[axiom_inst] assms by blast
438AOT_theorem "rule-ui:2[const_var]": assumes ‹∀α φ{α}› shows ‹φ{β}›
439  by (simp add: "rule-ui:1" "cqt:2[const_var]"[axiom_inst] assms)
440(* TODO: precise proviso in PLM *)
441AOT_theorem "rule-ui:2[lambda]":
442  assumes ‹∀F φ{F}› and ‹INSTANCE_OF_CQT_2(ψ)›
443  shows ‹φ{[λν1...νn ψ{ν1...νn}]}›
444  by (simp add: "rule-ui:1" "cqt:2[lambda]"[axiom_inst] assms)
445AOT_theorem "rule-ui:3": assumes ‹∀α φ{α}› shows ‹φ{α}›
446  by (simp add: "rule-ui:2[const_var]" assms)
447lemmas "∀E" = "rule-ui:1" "rule-ui:2[const_var]" "rule-ui:2[lambda]" "rule-ui:3"
448
449AOT_theorem "cqt-orig:1[const_var]": ‹∀α φ{α} → φ{β}› by (simp add: "∀E"(2) "→I")
450AOT_theorem "cqt-orig:1[lambda]":
451  assumes ‹INSTANCE_OF_CQT_2(ψ)›
452  shows ‹∀F φ{F} → φ{[λν1...νn ψ{ν1...νn}]}›
453  by (simp add: "∀E"(3) "→I" assms)
454AOT_theorem "cqt-orig:2": ‹∀α (φ → ψ{α}) → (φ → ∀α ψ{α})›
455  by (metis "→I" GEN "vdash-properties:6" "∀E"(4))
456AOT_theorem "cqt-orig:3": ‹∀α φ{α} → φ{α}› using "cqt-orig:1[const_var]" .
457
458(* TODO: work out difference to GEN *)
459AOT_theorem universal: assumes ‹for arbitrary β: φ{β}› shows ‹∀α φ{α}›
460  using GEN assms .
461lemmas "∀I" = universal
462
463(* Generalized mechanism for "∀I" followed by ∀E *)
464ML‹
465fun get_instantiated_allI ctxt varname thm = let
466val trm = Thm.concl_of thm
467val trm = case trm of (@{const Trueprop} $ (@{const AOT_model_valid_in} $ _ $ x)) => x
468                      | _ => raise Term.TERM ("Expected simple theorem.", [trm])
469fun extractVars (Const (const_name‹AOT_term_of_var›, _) $ Var v) =
470    (if fst (fst v) = fst varname then [Var v] else []) (* TODO: care about the index? *)
471  | extractVars (t1 $ t2) = extractVars t1 @ extractVars t2
472  | extractVars (Abs (_, _, t)) = extractVars t
473  | extractVars _ = []
474val vars = extractVars trm
475val vars = fold Term.add_vars vars []
476val var = hd vars
477val trmty = case (snd var) of (Type (type_name‹AOT_var›, [t])) => (t)
478              | _ => raise Term.TYPE ("Expected variable type.", [snd var], [Var var])
479val trm = Abs (Term.string_of_vname (fst var), trmty, Term.abstract_over (
480      Const (const_name‹AOT_term_of_var›, Type ("fun", [snd var, trmty]))
481       $ Var var, trm))
482val trm = Thm.cterm_of (Context.proof_of ctxt) trm
483val ty = hd (Term.add_tvars (Thm.prop_of @{thm "∀I"}) [])
484val typ = Thm.ctyp_of (Context.proof_of ctxt) trmty
485val allthm = Drule.instantiate_normalize ([(ty, typ)],[]) @{thm "∀I"}
486val phi = hd (Term.add_vars (Thm.prop_of allthm) [])
487val allthm = Drule.instantiate_normalize ([],[(phi,trm)]) allthm
488in
489allthm
490end
491›
492
493attribute_setup "∀I" =
494  ‹Scan.lift (Scan.repeat1 Args.var) >> (fn args => Thm.rule_attribute []
495  (fn ctxt => fn thm => fold (fn arg => fn thm => thm RS get_instantiated_allI ctxt arg thm) args thm))›
496  "Quantify over a variable in a theorem using GEN."
497
498attribute_setup "unvarify" =
499  ‹Scan.lift (Scan.repeat1 Args.var) >> (fn args => Thm.rule_attribute []
500  (fn ctxt => fn thm =>
501    let
502    val thm = fold (fn arg => fn thm => thm RS get_instantiated_allI ctxt arg thm) args thm
503    val thm = fold (fn _ => fn thm => thm RS @{thm "∀E"(1)}) args thm
504    in
505     thm
506    end))›
507  "Generalize a statement about variables to a statement about denoting terms."
508
509(* TODO: rereplace-lem does not apply to the embedding *)
510
511AOT_theorem "cqt-basic:1": ‹∀α∀β φ{α,β} ≡ ∀β∀α φ{α,β}›
512  by (metis "≡I" "∀E"(2) "∀I" "→I")
513
514AOT_theorem "cqt-basic:2": ‹∀α(φ{α} ≡ ψ{α}) ≡ (∀α(φ{α} → ψ{α}) & ∀α(ψ{α} → φ{α}))›
515proof (rule "≡I"; rule "→I")
516  AOT_assume ‹∀α(φ{α} ≡ ψ{α})›
517  AOT_hence ‹φ{α} ≡ ψ{α}› for α using "∀E"(2) by blast
518  AOT_hence ‹φ{α} → ψ{α}› and ‹ψ{α} → φ{α}› for α
519    using "≡E"(1,2) "→I" by blast+
520  AOT_thus ‹∀α(φ{α} → ψ{α}) & ∀α(ψ{α} → φ{α})›
521    by (auto intro: "&I" "∀I")
522next
523  AOT_assume ‹∀α(φ{α} → ψ{α}) & ∀α(ψ{α} → φ{α})›
524  AOT_hence ‹φ{α} → ψ{α}› and ‹ψ{α} → φ{α}› for α
525    using "∀E"(2) "&E" by blast+
526  AOT_hence ‹φ{α} ≡ ψ{α}› for α
527    using "≡I" by blast
528  AOT_thus ‹∀α(φ{α} ≡ ψ{α})› by (auto intro: "∀I")
529qed
530
531AOT_theorem "cqt-basic:3": ‹∀α(φ{α} ≡ ψ{α}) → (∀α φ{α} ≡ ∀α ψ{α})›
532proof(rule "→I")
533  AOT_assume ‹∀α(φ{α} ≡ ψ{α})›
534  AOT_hence 1: ‹φ{α} ≡ ψ{α}› for α using "∀E"(2) by blast
535  {
536    AOT_assume ‹∀α φ{α}›
537    AOT_hence ‹∀α ψ{α}› using 1 "∀I" "∀E"(4) "≡E" by metis
538  }
539  moreover {
540    AOT_assume ‹∀α ψ{α}›
541    AOT_hence ‹∀α φ{α}› using 1 "∀I" "∀E"(4) "≡E" by metis
542  }
543  ultimately AOT_show ‹∀α φ{α} ≡ ∀α ψ{α}›
544    using "≡I" "→I" by auto
545qed
546
547AOT_theorem "cqt-basic:4": ‹∀α(φ{α} & ψ{α}) → (∀α φ{α} & ∀α ψ{α})›
548proof(rule "→I")
549  AOT_assume 0: ‹∀α(φ{α} & ψ{α})›
550  AOT_have ‹φ{α}› and ‹ψ{α}› for α using "∀E"(2) 0 "&E" by blast+
551  AOT_thus ‹∀α φ{α} & ∀α ψ{α}›
552    by (auto intro: "∀I" "&I")
553qed
554
555AOT_theorem "cqt-basic:5": ‹(∀α1...∀αn(φ{α1...αn})) → φ{α1...αn}›
556  using "cqt-orig:3" by blast
557
558AOT_theorem "cqt-basic:6": ‹∀α∀α φ{α} ≡ ∀α φ{α}›
559  by (meson "≡I" "→I" GEN "cqt-orig:1[const_var]")
560
561AOT_theorem "cqt-basic:7": ‹(φ → ∀α ψ{α}) ≡ ∀α(φ → ψ{α})›
562  by (metis "→I" "vdash-properties:6" "rule-ui:3" "≡I" GEN)
563
564AOT_theorem "cqt-basic:8": ‹(∀α φ{α} ∨ ∀α ψ{α}) → ∀α (φ{α} ∨ ψ{α})›
565  by (simp add: "∨I"(3) "→I" GEN "cqt-orig:1[const_var]")
566
567AOT_theorem "cqt-basic:9": ‹(∀α (φ{α} → ψ{α}) & ∀α (ψ{α} → χ{α})) → ∀α(φ{α} → χ{α})›
568proof -
569  {
570    AOT_assume ‹∀α (φ{α} → ψ{α})›
571    moreover AOT_assume ‹∀α (ψ{α} → χ{α})›
572    ultimately AOT_have ‹φ{α} → ψ{α}› and ‹ψ{α} → χ{α}› for α using "∀E" by blast+
573    AOT_hence ‹φ{α} → χ{α}› for α by (metis "→E" "→I")
574    AOT_hence ‹∀α(φ{α} → χ{α})› using "∀I" by fast
575  }
576  thus ?thesis using "&I" "→I" "&E" by meson
577qed
578
579AOT_theorem "cqt-basic:10": ‹(∀α(φ{α} ≡ ψ{α}) & ∀α(ψ{α} ≡ χ{α})) → ∀α (φ{α} ≡ χ{α})›
580proof(rule "→I"; rule "∀I")
581  fix β
582  AOT_assume ‹∀α(φ{α} ≡ ψ{α}) & ∀α(ψ{α} ≡ χ{α})›
583  AOT_hence ‹φ{β} ≡ ψ{β}› and ‹ψ{β} ≡ χ{β}› using "&E" "∀E" by blast+
584  AOT_thus ‹φ{β} ≡ χ{β}› using "≡I" "≡E" by blast
585qed
586
587AOT_theorem "cqt-basic:11": ‹∀α(φ{α} ≡ ψ{α}) ≡ ∀α (ψ{α} ≡ φ{α})›
588proof (rule "≡I"; rule "→I")
589  AOT_assume 0: ‹∀α(φ{α} ≡ ψ{α})›
590  {
591    fix α
592    AOT_have ‹φ{α} ≡ ψ{α}› using 0 "∀E" by blast
593    AOT_hence ‹ψ{α} ≡ φ{α}› using "≡I" "≡E" "→I" "→E" by metis
594  }
595  AOT_thus ‹∀α(ψ{α} ≡ φ{α})› using "∀I" by fast
596next
597  AOT_assume 0: ‹∀α(ψ{α} ≡ φ{α})›
598  {
599    fix α
600    AOT_have ‹ψ{α} ≡ φ{α}› using 0 "∀E" by blast
601    AOT_hence ‹φ{α} ≡ ψ{α}› using "≡I" "≡E" "→I" "→E" by metis
602  }
603  AOT_thus ‹∀α(φ{α} ≡ ψ{α})› using "∀I" by fast
604qed
605
606AOT_theorem "cqt-basic:12": ‹∀α φ{α} → ∀α (ψ{α} → φ{α})›
607  by (simp add: "∀E"(2) "→I" GEN)
608
609AOT_theorem "cqt-basic:13": ‹∀α φ{α} ≡ ∀β φ{β}›
610  using "≡I" "→I" by blast
611
612AOT_theorem "cqt-basic:14": ‹(∀α1...∀αn (φ{α1...αn} → ψ{α1...αn})) → ((∀α1...∀αn φ{α1...αn}) → (∀α1...∀αn ψ{α1...αn}))›
613  using "cqt:3"[axiom_inst] by auto
614
615AOT_theorem "cqt-basic:15": ‹(∀α1...∀αn (φ → ψ{α1...αn})) → (φ → (∀α1...∀αn ψ{α1...αn}))›
616  using "cqt-orig:2" by auto
617
618(* TODO: once more the same in the embedding... need to distinguish these better *)
619AOT_theorem "universal-cor": assumes ‹for arbitrary β: φ{β}›  shows ‹∀α φ{α}›
620  using GEN assms .
621
622AOT_theorem "existential:1": assumes ‹φ{τ}› and ‹τ↓› shows ‹∃α φ{α}›
623proof(rule "raa-cor:1")
624  AOT_assume ‹¬∃α φ{α}›
625  AOT_hence ‹∀α ¬φ{α}›
626    using "≡dfI" "conventions:4" RAA "&I" by blast
627  AOT_hence ‹¬φ{τ}› using assms(2) "∀E"(1) "→E" by blast
628  AOT_thus ‹φ{τ} & ¬φ{τ}› using assms(1) "&I" by blast
629qed
630
631AOT_theorem "existential:2[const_var]": assumes ‹φ{β}› shows ‹∃α φ{α}›
632  using "existential:1" "cqt:2[const_var]"[axiom_inst] assms by blast
633
634AOT_theorem "existential:2[lambda]":
635  assumes ‹φ{[λν1...νn ψ{ν1...νn}]}› and ‹INSTANCE_OF_CQT_2(ψ)›
636  shows ‹∃α φ{α}›
637  using "existential:1" "cqt:2[lambda]"[axiom_inst] assms by blast
638lemmas "∃I" = "existential:1" "existential:2[const_var]" "existential:2[lambda]" 
639
640AOT_theorem "instantiation":
641  assumes ‹for arbitrary β: φ{β} ⊢ ψ› and ‹∃α φ{α}›
642  shows ‹ψ›
643  by (metis (no_types, lifting) "≡dfE" GEN "raa-cor:3" "conventions:4" assms)
644lemmas "∃E" = "instantiation"
645
646AOT_theorem "cqt-further:1": ‹∀α φ{α} → ∃α φ{α}›
647  using "∀E"(4) "∃I"(2) "→I" by metis
648
649AOT_theorem "cqt-further:2": ‹¬∀α φ{α} → ∃α ¬φ{α}›
650  using "∀I" "∃I"(2) "→I" RAA by metis
651
652AOT_theorem "cqt-further:3": ‹∀α φ{α} ≡ ¬∃α ¬φ{α}›
653  using "∀E"(4) "∃E" "→I" RAA
654  by (metis "cqt-further:2" "≡I" "modus-tollens:1")
655
656AOT_theorem "cqt-further:4": ‹¬∃α φ{α} → ∀α ¬φ{α}›
657  using "∀I" "∃I"(2)"→I" RAA by metis
658
659AOT_theorem "cqt-further:5": ‹∃α (φ{α} & ψ{α}) → (∃α φ{α} & ∃α ψ{α})›
660  by (metis (no_types, lifting) "&E" "&I" "∃E" "∃I"(2) "→I")
661
662AOT_theorem "cqt-further:6": ‹∃α (φ{α} ∨ ψ{α}) → (∃α φ{α} ∨ ∃α ψ{α})›
663  by (metis (mono_tags, lifting) "∃E" "∃I"(2) "∨E"(3) "∨I"(1, 2) "→I" RAA(2))
664
665AOT_theorem "cqt-further:7": ‹∃α φ{α} ≡ ∃β φ{β}› (* TODO: vacuous in the embedding *)
666  by (simp add: "oth-class-taut:3:a")
667
668AOT_theorem "cqt-further:8": ‹(∀α φ{α} & ∀α ψ{α}) → ∀α (φ{α} ≡ ψ{α})›
669  by (metis (mono_tags, lifting) "&E" "≡I" "∀E"(2) "→I" GEN)
670
671AOT_theorem "cqt-further:9": ‹(¬∃α φ{α} & ¬∃α ψ{α}) → ∀α (φ{α} ≡ ψ{α})›
672  by (metis (mono_tags, lifting) "&E" "≡I" "∃I"(2) "→I" GEN "raa-cor:4")
673
674AOT_theorem "cqt-further:10": ‹(∃α φ{α} & ¬∃α ψ{α}) → ¬∀α (φ{α} ≡ ψ{α})›
675proof(rule "→I"; rule "raa-cor:2")
676  AOT_assume 0: ‹∃α φ{α} & ¬∃α ψ{α}›
677  then AOT_obtain α where ‹φ{α}› using "∃E" "&E"(1) by metis
678  moreover AOT_assume ‹∀α (φ{α} ≡ ψ{α})›
679  ultimately AOT_have ‹ψ{α}› using "∀E"(4) "≡E"(1) by blast
680  AOT_hence ‹∃α ψ{α}› using "∃I" by blast
681  AOT_thus ‹∃α ψ{α} & ¬∃α ψ{α}› using 0 "&E"(2) "&I" by blast
682qed
683
684AOT_theorem "cqt-further:11": ‹∃α∃β φ{α,β} ≡ ∃β∃α φ{α,β}›
685  using "≡I" "→I" "∃I"(2) "∃E" by metis
686
687AOT_theorem "log-prop-prop:1": ‹[λ φ]↓›
688  using "cqt:2[lambda0]"[axiom_inst] by auto
689
690AOT_theorem "log-prop-prop:2": ‹φ↓›
691  by (rule "≡dfI"[OF "existence:3"]) "cqt:2[lambda]"
692
693AOT_theorem "exist-nec": ‹τ↓ → □τ↓›
694proof -
695  AOT_have ‹∀β □β↓›
696    by (simp add: GEN RN "cqt:2[const_var]"[axiom_inst])
697  AOT_thus ‹τ↓ → □τ↓›
698    using "cqt:1"[axiom_inst] "→E" by blast
699qed
700
701(* TODO: replace this mechanism by a "proof by types" command *)
702class AOT_Term_id = AOT_Term +
703  assumes "t=t-proper:1"[AOT]: ‹[v ⊨ τ = τ' → τ↓]›
704      and "t=t-proper:2"[AOT]: ‹[v ⊨ τ = τ' → τ'↓]›
705
706instance κ :: AOT_Term_id
707proof
708  AOT_modally_strict {
709    AOT_show ‹κ = κ' → κ↓› for κ κ'
710    proof(rule "→I")
711      AOT_assume ‹κ = κ'›
712      AOT_hence ‹O!κ ∨ A!κ›
713        by (rule "∨I"(3)[OF "≡dfE"[OF "identity:1"]])
714           (meson "→I" "∨I"(1) "&E"(1))+
715      AOT_thus ‹κ↓›
716        by (rule "∨E"(1))
717           (metis "cqt:5:a"[axiom_inst] "→I" "→E" "&E"(2))+
718    qed
719  }
720next
721  AOT_modally_strict {
722    AOT_show ‹κ = κ' → κ'↓› for κ κ'
723    proof(rule "→I")
724      AOT_assume ‹κ = κ'›
725      AOT_hence ‹O!κ' ∨ A!κ'›
726        by (rule "∨I"(3)[OF "≡dfE"[OF "identity:1"]])
727           (meson "→I" "∨I" "&E")+
728      AOT_thus ‹κ'↓›
729        by (rule "∨E"(1))
730           (metis "cqt:5:a"[axiom_inst] "→I" "→E" "&E"(2))+
731    qed
732  }
733qed
734
735instance rel :: (AOT_κs) AOT_Term_id
736proof
737  AOT_modally_strict {
738    AOT_show ‹Π = Π' → Π↓› for Π Π' :: ‹<'a>› (* TODO: how to get rid of the fixes? *)
739    proof(rule "→I")
740      AOT_assume ‹Π = Π'›
741      AOT_thus ‹Π↓› using "≡dfE"[OF "identity:3"[of Π Π']] "&E" by blast
742    qed
743  }
744next
745  AOT_modally_strict {
746    AOT_show ‹Π = Π' → Π'↓› for Π Π' :: ‹<'a>› (* TODO: how to get rid of the fixes? *)
747    proof(rule "→I")
748      AOT_assume ‹Π = Π'›
749      AOT_thus ‹Π'↓› using "≡dfE"[OF "identity:3"[of Π Π']] "&E" by blast
750    qed
751  }
752qed
753
754instance 𝗈 :: AOT_Term_id
755proof
756  AOT_modally_strict {
757    fix φ ψ
758    AOT_show ‹φ = ψ → φ↓›
759    proof(rule "→I")
760      AOT_assume ‹φ = ψ›
761      AOT_thus ‹φ↓› using "≡dfE"[OF "identity:4"[of φ ψ]] "&E" by blast
762    qed
763  }
764next
765  AOT_modally_strict {
766    fix φ ψ
767    AOT_show ‹φ = ψ → ψ↓›
768    proof(rule "→I")
769      AOT_assume ‹φ = ψ›
770      AOT_thus ‹ψ↓› using "≡dfE"[OF "identity:4"[of φ ψ]] "&E" by blast
771    qed
772  }
773qed
774
775instance prod :: (AOT_Term_id, AOT_Term_id) AOT_Term_id
776proof
777  AOT_modally_strict {
778    fix τ τ' :: ‹'a×'b›
779    AOT_show ‹τ = τ' → τ↓›
780    proof (induct τ; induct τ'; rule "→I")
781      fix τ1 τ1' :: 'a and τ2  τ2' :: 'b
782      AOT_assume ‹«(τ1, τ2)» = «(τ1', τ2')»›
783      AOT_hence ‹(τ1 = τ1') & (τ2 = τ2')› by (metis "≡dfE" tuple_identity_1)
784      AOT_hence ‹τ1↓› and ‹τ2↓› using "t=t-proper:1" "&E" "vdash-properties:10" by blast+
785      AOT_thus ‹«(τ1, τ2)»↓› by (metis "≡dfI" "&I" tuple_denotes)
786    qed
787  }
788next
789  AOT_modally_strict {
790    fix τ τ' :: ‹'a×'b›
791    AOT_show ‹τ = τ' → τ'↓›
792    proof (induct τ; induct τ'; rule "→I")
793      fix τ1 τ1' :: 'a and τ2  τ2' :: 'b
794      AOT_assume ‹«(τ1, τ2)» = «(τ1', τ2')»›
795      AOT_hence ‹(τ1 = τ1') & (τ2 = τ2')› by (metis "≡dfE" tuple_identity_1)
796      AOT_hence ‹τ1'↓› and ‹τ2'↓› using "t=t-proper:2" "&E" "vdash-properties:10" by blast+
797      AOT_thus ‹«(τ1', τ2')»↓› by (metis "≡dfI" "&I" tuple_denotes)
798    qed
799  }
800qed
801
802(* TODO: this is the end of the "proof by types" and makes the results available on new theorems *)
803AOT_register_type_constraints
804  Term: ‹_::AOT_Term_id› ‹_::AOT_Term_id›
805AOT_register_type_constraints
806  Individual: ‹κ› ‹_::{AOT_κs, AOT_Term_id}›
807AOT_register_type_constraints
808  Relation: ‹<_::{AOT_κs, AOT_Term_id}>›
809
810AOT_theorem "id-rel-nec-equiv:1": ‹Π = Π' → □∀x1...∀xn ([Π]x1...xn ≡ [Π']x1...xn)›
811proof(rule "→I")
812  AOT_assume assumption: ‹Π = Π'›
813  AOT_hence ‹Π↓› and ‹Π'↓›
814    using "t=t-proper:1" "t=t-proper:2" MP by blast+
815  moreover AOT_have ‹∀F∀G (F = G → ((□∀x1...∀xn ([F]x1...xn ≡ [F]x1...xn)) → □∀x1...∀xn ([F]x1...xn ≡ [G]x1...xn)))›
816    apply (rule GEN)+ using "l-identity"[axiom_inst] by force
817  ultimately AOT_have ‹Π = Π' → ((□∀x1...∀xn ([Π]x1...xn ≡ [Π]x1...xn)) → □∀x1...∀xn ([Π]x1...xn ≡ [Π']x1...xn))›
818    using "∀E"(1) by blast
819  AOT_hence ‹(□∀x1...∀xn ([Π]x1...xn ≡ [Π]x1...xn)) → □∀x1...∀xn ([Π]x1...xn ≡ [Π']x1...xn)›
820    using assumption "→E" by blast
821  moreover AOT_have ‹□∀x1...∀xn ([Π]x1...xn ≡ [Π]x1...xn)›
822    by (simp add: RN "oth-class-taut:3:a" "universal-cor")
823  ultimately AOT_show ‹□∀x1...∀xn ([Π]x1...xn ≡ [Π']x1...xn)›
824    using "→E" by blast
825qed
826
827AOT_theorem "id-rel-nec-equiv:2": ‹φ = ψ → □(φ ≡ ψ)›
828proof(rule "→I")
829  AOT_assume assumption: ‹φ = ψ›
830  AOT_hence ‹φ↓› and ‹ψ↓›
831    using "t=t-proper:1" "t=t-proper:2" MP by blast+
832  moreover AOT_have ‹∀p∀q (p = q → ((□(p ≡ p) → □(p ≡ q))))›
833    apply (rule GEN)+ using "l-identity"[axiom_inst] by force
834  ultimately AOT_have ‹φ = ψ → (□(φ ≡ φ) → □(φ ≡ ψ))›
835    using "∀E"(1) by blast
836  AOT_hence ‹□(φ ≡ φ) → □(φ ≡ ψ)›
837    using assumption "→E" by blast
838  moreover AOT_have ‹□(φ ≡ φ)›
839    by (simp add: RN "oth-class-taut:3:a" "universal-cor")
840  ultimately AOT_show ‹□(φ ≡ ψ)›
841    using "→E" by blast
842qed
843
844AOT_theorem "rule=E": assumes ‹φ{τ}› and ‹τ = σ› shows ‹φ{σ}›
845proof -
846  AOT_have ‹τ↓› and ‹σ↓› using assms(2) "t=t-proper:1" "t=t-proper:2" "→E" by blast+
847  moreover AOT_have ‹∀α∀β(α = β → (φ{α} → φ{β}))›
848    apply (rule GEN)+ using "l-identity"[axiom_inst] by blast
849  ultimately AOT_have ‹τ = σ → (φ{τ} → φ{σ})›
850    using "∀E"(1) by blast
851  AOT_thus ‹φ{σ}› using assms "→E" by blast
852qed
853
854AOT_theorem "propositions-lemma:1": ‹[λ φ] = φ›
855proof -
856  AOT_have ‹φ↓› by (simp add: "log-prop-prop:2")
857  moreover AOT_have ‹∀p [λ p] = p› using "lambda-predicates:3[zero]"[axiom_inst] "∀I" by fast
858  ultimately AOT_show ‹[λ φ] = φ›
859    using "∀E" by blast
860qed
861
862AOT_theorem "propositions-lemma:2": ‹[λ φ] ≡ φ›
863proof -
864  AOT_have ‹[λ φ] ≡ [λ φ]› by (simp add: "oth-class-taut:3:a")
865  AOT_thus ‹[λ φ] ≡ φ› using "propositions-lemma:1" "rule=E" by blast
866qed
867
868(* propositions-lemma:3 through propositions-lemma:5 do not apply *)
869
870AOT_theorem "propositions-lemma:6": ‹(φ ≡ ψ) ≡ ([λ φ] ≡ [λ ψ])›
871  by (metis "≡E"(1) "≡E"(5) "Associativity of ≡" "propositions-lemma:2")
872
873(* dr-alphabetic-rules does not apply *)
874
875AOT_theorem "oa-exist:1": ‹O!↓›
876proof -
877  AOT_have ‹[λx ◇[E!]x]↓› by "cqt:2[lambda]"
878  AOT_hence 1: ‹O! = [λx ◇[E!]x]› using "df-rules-terms[4]"[OF "oa:1", THEN "&E"(1)] "→E" by blast
879  AOT_show ‹O!↓› using "t=t-proper:1"[THEN "→E", OF 1] by simp
880qed
881
882AOT_theorem "oa-exist:2": ‹A!↓›
883proof -
884  AOT_have ‹[λx ¬◇[E!]x]↓› by "cqt:2[lambda]"
885  AOT_hence 1: ‹A! = [λx ¬◇[E!]x]› using "df-rules-terms[4]"[OF "oa:2", THEN "&E"(1)] "→E" by blast
886  AOT_show ‹A!↓› using "t=t-proper:1"[THEN "→E", OF 1] by simp
887qed
888
889AOT_theorem "oa-exist:3": ‹O!x ∨ A!x›
890proof(rule "raa-cor:1")
891  AOT_assume ‹¬(O!x ∨ A!x)›
892  AOT_hence A: ‹¬O!x› and B: ‹¬A!x›
893    using "Disjunction Addition"(1) "modus-tollens:1" "∨I"(2) "raa-cor:5" by blast+
894  AOT_have C: ‹O! = [λx ◇[E!]x]›
895    by (rule "df-rules-terms[4]"[OF "oa:1", THEN "&E"(1), THEN "→E"]) "cqt:2[lambda]"
896  AOT_have D: ‹A! = [λx ¬◇[E!]x]›
897    by (rule "df-rules-terms[4]"[OF "oa:2", THEN "&E"(1), THEN "→E"]) "cqt:2[lambda]"
898  AOT_have E: ‹¬[λx ◇[E!]x]x›
899    using A C "rule=E" by fast
900  AOT_have F: ‹¬[λx ¬◇[E!]x]x›
901    using B D "rule=E" by fast
902  AOT_have G: ‹[λx ◇[E!]x]x ≡ ◇[E!]x›
903    by (rule "lambda-predicates:2"[axiom_inst, THEN "→E"]) "cqt:2[lambda]"
904  AOT_have H: ‹[λx ¬◇[E!]x]x ≡ ¬◇[E!]x›
905    by (rule "lambda-predicates:2"[axiom_inst, THEN "→E"]) "cqt:2[lambda]"
906  AOT_show ‹¬◇[E!]x & ¬¬◇[E!]x› using G E "≡E" H F "≡E" "&I" by metis
907qed
908
909AOT_theorem "p-identity-thm2:1": ‹F = G ≡ □∀x(x[F] ≡ x[G])›
910proof -
911  AOT_have ‹F = G ≡ F↓ & G↓ & □∀x(x[F] ≡ x[G])›
912    using "identity:2" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
913  moreover AOT_have ‹F↓› and ‹G↓›
914    by (auto simp: "cqt:2[const_var]"[axiom_inst])
915  ultimately AOT_show ‹F = G ≡ □∀x(x[F] ≡ x[G])›
916    using "≡S"(1) "&I" by blast
917qed
918
919AOT_theorem "p-identity-thm2:2[2]": ‹F = G ≡ ∀y1([λx [F]xy1] = [λx [G]xy1] & [λx [F]y1x] = [λx [G]y1x])›
920proof -
921  AOT_have ‹F = G ≡ F↓ & G↓ & ∀y1([λx [F]xy1] = [λx [G]xy1] & [λx [F]y1x] = [λx [G]y1x])›
922    using "identity:3[2]" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
923  moreover AOT_have ‹F↓› and ‹G↓›
924    by (auto simp: "cqt:2[const_var]"[axiom_inst])
925  ultimately show ?thesis
926    using "≡S"(1) "&I" by blast
927qed
928    
929AOT_theorem "p-identity-thm2:2[3]": ‹F = G ≡ ∀y1∀y2([λx [F]xy1y2] = [λx [G]xy1y2] & [λx [F]y1xy2] = [λx [G]y1xy2] & [λx [F]y1y2x] = [λx [G]y1y2x])›
930proof -
931  AOT_have ‹F = G ≡ F↓ & G↓ & ∀y1∀y2([λx [F]xy1y2] = [λx [G]xy1y2] & [λx [F]y1xy2] = [λx [G]y1xy2] & [λx [F]y1y2x] = [λx [G]y1y2x])›
932    using "identity:3[3]" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
933  moreover AOT_have ‹F↓› and ‹G↓›
934    by (auto simp: "cqt:2[const_var]"[axiom_inst])
935  ultimately show ?thesis
936    using "≡S"(1) "&I" by blast
937qed
938
939AOT_theorem "p-identity-thm2:2[4]": ‹F = G ≡ ∀y1∀y2∀y3([λx [F]xy1y2y3] = [λx [G]xy1y2y3] & [λx [F]y1xy2y3] = [λx [G]y1xy2y3] & [λx [F]y1y2xy3] = [λx [G]y1y2xy3] & [λx [F]y1y2y3x] = [λx [G]y1y2y3x])›
940proof -
941  AOT_have ‹F = G ≡ F↓ & G↓ & ∀y1∀y2∀y3([λx [F]xy1y2y3] = [λx [G]xy1y2y3] & [λx [F]y1xy2y3] = [λx [G]y1xy2y3] & [λx [F]y1y2xy3] = [λx [G]y1y2xy3] & [λx [F]y1y2y3x] = [λx [G]y1y2y3x])›
942    using "identity:3[4]" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
943  moreover AOT_have ‹F↓› and ‹G↓›
944    by (auto simp: "cqt:2[const_var]"[axiom_inst])
945  ultimately show ?thesis
946    using "≡S"(1) "&I" by blast
947qed
948
949AOT_theorem "p-identity-thm2:2":
950  ‹F = G ≡ ∀x1...∀xn «AOT_sem_proj_id x1xn (λ τ . «[F]τ») (λ τ . «[G]τ»)»›
951proof -
952  AOT_have ‹F = G ≡ F↓ & G↓ & ∀x1...∀xn «AOT_sem_proj_id x1xn (λ τ . «[F]τ») (λ τ . «[G]τ»)»›
953    using "identity:3" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
954  moreover AOT_have ‹F↓› and ‹G↓›
955    by (auto simp: "cqt:2[const_var]"[axiom_inst])
956  ultimately show ?thesis
957    using "≡S"(1) "&I" by blast
958qed
959
960AOT_theorem "p-identity-thm2:3":
961  ‹p = q ≡ [λx p] = [λx q]›
962proof -
963  AOT_have ‹p = q ≡ p↓ & q↓ & [λx p] = [λx q]›
964    using "identity:4" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
965  moreover AOT_have ‹p↓› and ‹q↓›
966    by (auto simp: "cqt:2[const_var]"[axiom_inst])
967  ultimately show ?thesis
968    using "≡S"(1) "&I" by blast
969qed
970
971class AOT_Term_id_2 = AOT_Term_id + assumes "id-eq:1": ‹[v ⊨ α = α]›
972
973instance κ :: AOT_Term_id_2
974proof
975  AOT_modally_strict {
976    fix x
977    {
978      AOT_assume ‹O!x›
979      moreover AOT_have ‹□∀F([F]x ≡ [F]x)›
980        using RN GEN "oth-class-taut:3:a" by fast
981      ultimately AOT_have ‹O!x & O!x & □∀F([F]x ≡ [F]x)› using "&I" by simp
982    }
983    moreover {
984      AOT_assume ‹A!x›
985      moreover AOT_have ‹□∀F(x[F] ≡ x[F])›
986        using RN GEN "oth-class-taut:3:a" by fast
987      ultimately AOT_have ‹A!x & A!x & □∀F(x[F] ≡ x[F])› using "&I" by simp
988    }
989    ultimately AOT_have ‹(O!x & O!x & □∀F([F]x ≡ [F]x)) ∨ (A!x & A!x & □∀F(x[F] ≡ x[F]))›
990      using "oa-exist:3" "∨I"(1) "∨I"(2) "∨E"(3) "raa-cor:1" by blast
991    AOT_thus ‹x = x›
992      using "identity:1"[THEN "df-rules-formulas[4]"] "→E" by blast
993  }
994qed
995
996instance rel :: ("{AOT_κs,AOT_Term_id_2}") AOT_Term_id_2
997proof
998  AOT_modally_strict {
999    fix F :: "<'a> AOT_var"
1000    AOT_have 0: ‹[λx1...xn [F]x1...xn] = F›
1001      by (simp add: "lambda-predicates:3"[axiom_inst])
1002    AOT_have ‹[λx1...xn [F]x1...xn]↓›
1003      by "cqt:2[lambda]"
1004    AOT_hence ‹[λx1...xn [F]x1...xn] = [λx1...xn [F]x1...xn]›
1005      using "lambda-predicates:1"[axiom_inst] "→E" by blast
1006    AOT_show ‹F = F› using "rule=E" 0 by force 
1007  }
1008qed
1009
1010instance 𝗈 :: AOT_Term_id_2
1011proof
1012  AOT_modally_strict {
1013    fix p
1014    AOT_have 0: ‹[λ p] = p›
1015      by (simp add: "lambda-predicates:3[zero]"[axiom_inst])
1016    AOT_have ‹[λ p]↓›
1017      by (rule "cqt:2[lambda0]"[axiom_inst])
1018    AOT_hence ‹[λ p] = [λ p]›
1019      using "lambda-predicates:1[zero]"[axiom_inst] "→E" by blast
1020    AOT_show ‹p = p› using "rule=E" 0 by force
1021  }
1022qed
1023
1024instance prod :: (AOT_Term_id_2, AOT_Term_id_2) AOT_Term_id_2
1025proof
1026  AOT_modally_strict {
1027    fix α :: ‹('a×'b) AOT_var›
1028    AOT_show ‹α = α›
1029    proof (induct)
1030      AOT_show ‹τ = τ› if ‹τ↓› for τ :: ‹'a×'b›
1031        using that
1032      proof (induct τ)
1033        fix τ1 :: 'a and τ2 :: 'b
1034        AOT_assume ‹«(τ1,τ2)»↓›
1035        AOT_hence ‹τ1↓› and ‹τ2↓› using "≡dfE" "&E" tuple_denotes by blast+
1036        AOT_hence ‹τ1 = τ1› and ‹τ2 = τ2› using "id-eq:1"[unvarify α] by blast+
1037        AOT_thus ‹«(τ1, τ2)» = «(τ1, τ2)»› by (metis "≡dfI" "&I" tuple_identity_1)
1038      qed
1039    qed
1040  }
1041qed
1042
1043AOT_register_type_constraints
1044  Term: ‹_::AOT_Term_id_2› ‹_::AOT_Term_id_2›
1045AOT_register_type_constraints
1046  Individual: ‹κ› ‹_::{AOT_κs, AOT_Term_id_2}›
1047AOT_register_type_constraints
1048  Relation: ‹<_::{AOT_κs, AOT_Term_id_2}>›
1049
1050(* TODO: Interestingly, this doesn't depend on "id-eq:1" at all! *)
1051AOT_theorem "id-eq:2": ‹α = β → β = α›
1052(*
1053  TODO: look at this proof generated using:
1054        including AOT_no_atp sledgehammer[isar_proofs = true]
1055proof -
1056  have "(∃φ. [v ⊨ ~β = α → ~φ] ∧ [v ⊨ α = β → φ]) ∨ (∃φ. ¬ [v ⊨ φ{α} → φ{β}])"
1057    by meson
1058  then show ?thesis
1059    by (meson "contraposition:2" "Hypothetical Syllogism" "deduction-theorem" l_"identity:1" "useful-tautologies:1")
1060qed
1061*)
1062(*  by (meson "rule=E" "deduction-theorem") *)
1063proof (rule "→I")
1064  AOT_assume ‹α = β›
1065  moreover AOT_have ‹β = β› using calculation "rule=E"[of _ "λ τ . «τ = β»" "AOT_term_of_var α" "AOT_term_of_var β"] by blast
1066  moreover AOT_have ‹α = α → α = α› using "if-p-then-p" by blast
1067  ultimately AOT_show ‹β = α›
1068    using "→E" "→I" "rule=E"[of _ "λ τ . «(τ = τ) → (τ = α)»" "AOT_term_of_var α" "AOT_term_of_var β"] by blast
1069qed
1070
1071AOT_theorem "id-eq:3": ‹α = β & β = γ → α = γ›
1072  using "rule=E" "→I" "&E" by blast
1073
1074AOT_theorem "id-eq:4": ‹α = β ≡ ∀γ (α = γ ≡ β = γ)›
1075proof (rule "≡I"; rule "→I")
1076  AOT_assume 0: ‹α = β›
1077  AOT_hence 1: ‹β = α› using "id-eq:2" "→E" by blast
1078  AOT_show ‹∀γ (α = γ ≡ β = γ)›
1079    by (rule GEN) (metis "≡I" "→I" 0 "1" "rule=E")
1080next
1081  AOT_assume ‹∀γ (α = γ ≡ β = γ)›
1082  AOT_hence ‹α = α ≡ β = α› using "∀E"(2) by blast
1083  AOT_hence ‹α = α → β = α› using "≡E"(1) "→I" by blast
1084  AOT_hence ‹β = α› using "id-eq:1" "→E" by blast
1085  AOT_thus ‹α = β› using "id-eq:2" "→E" by blast
1086qed
1087
1088AOT_theorem "rule=I:1": assumes ‹τ↓› shows ‹τ = τ›
1089proof -
1090  AOT_have ‹∀α (α = α)›
1091    by (rule GEN) (metis "id-eq:1")
1092  AOT_thus ‹τ = τ› using assms "∀E" by blast
1093qed
1094
1095AOT_theorem "rule=I:2[const_var]": "α = α"
1096  using "id-eq:1".
1097
1098AOT_theorem "rule=I:2[lambda]": assumes ‹INSTANCE_OF_CQT_2(φ)› shows "[λν1...νn φ{ν1...νn}] = [λν1...νn φ{ν1...νn}]"
1099proof -
1100  AOT_have ‹∀α (α = α)›
1101    by (rule GEN) (metis "id-eq:1")
1102  moreover AOT_have ‹[λν1...νn φ{ν1...νn}]↓› using assms by (rule "cqt:2[lambda]"[axiom_inst])
1103  ultimately AOT_show ‹[λν1...νn φ{ν1...νn}] = [λν1...νn φ{ν1...νn}]› using assms "∀E" by blast
1104qed
1105
1106lemmas "=I" = "rule=I:1" "rule=I:2[const_var]" "rule=I:2[lambda]"
1107
1108AOT_theorem "rule-id-df:1":
1109  assumes ‹τ{α1...αn} =df σ{α1...αn}› and ‹σ{τ1...τn}↓›
1110  shows ‹τ{τ1...τn} = σ{τ1...τn}›
1111proof -
1112  AOT_have ‹σ{τ1...τn}↓ → τ{τ1...τn} = σ{τ1...τn}›
1113    using "df-rules-terms[3]" assms(1) "&E" by blast
1114  AOT_thus ‹τ{τ1...τn} = σ{τ1...τn}›
1115    using assms(2) "→E" by blast
1116qed
1117
1118AOT_theorem "rule-id-df:1[zero]":
1119  assumes ‹τ =df σ› and ‹σ↓›
1120  shows ‹τ = σ›
1121proof -
1122  AOT_have ‹σ↓ → τ = σ›
1123    using "df-rules-terms[4]" assms(1) "&E" by blast
1124  AOT_thus ‹τ = σ›
1125    using assms(2) "→E" by blast
1126qed
1127
1128AOT_theorem "rule-id-df:2:a":
1129  assumes ‹τ{α1...αn} =df σ{α1...αn}› and ‹σ{τ1...τn}↓› and ‹φ{τ{τ1...τn}}›
1130  shows ‹φ{σ{τ1...τn}}›
1131proof -
1132  AOT_have ‹τ{τ1...τn} = σ{τ1...τn}› using "rule-id-df:1" assms(1,2) by blast
1133  AOT_thus ‹φ{σ{τ1...τn}}› using assms(3) "rule=E" by blast
1134qed
1135
1136(* TODO: get rid of this, ideally *)
1137AOT_theorem "rule-id-df:2:a[2]":
1138  assumes ‹τ{«(α1,α2)»} =df σ{«(α1,α2)»}› and ‹σ{«(τ1,τ2)»}↓› and ‹φ{τ{«(τ1,τ2)»}}›
1139  shows ‹φ{σ{«(τ1,τ2)»}}›
1140proof -
1141  AOT_have ‹τ{«(τ1,τ2)»} = σ{«(τ1,τ2)»}›
1142  proof -
1143    AOT_have ‹σ{«(τ1,τ2)»}↓ → τ{«(τ1,τ2)»} = σ{«(τ1,τ2)»}›
1144      using assms by (simp add: AOT_sem_conj AOT_sem_imp AOT_sem_eq AOT_sem_not AOT_sem_denotes AOT_model_id_def) (* NOTE: semantics needed *)
1145    AOT_thus ‹τ{«(τ1,τ2)»} = σ{«(τ1,τ2)»}›
1146      using assms(2) "→E" by blast
1147  qed
1148  AOT_thus ‹φ{σ{«(τ1,τ2)»}}› using assms(3) "rule=E" by blast
1149qed
1150
1151AOT_theorem "rule-id-df:2:a[zero]":
1152  assumes ‹τ =df σ› and ‹σ↓› and ‹φ{τ}›
1153  shows ‹φ{σ}›
1154proof -
1155  AOT_have ‹τ = σ› using "rule-id-df:1[zero]" assms(1,2) by blast
1156  AOT_thus ‹φ{σ}› using assms(3) "rule=E" by blast
1157qed
1158
1159lemmas "=dfE" = "rule-id-df:2:a" "rule-id-df:2:a[zero]"
1160
1161AOT_theorem "rule-id-df:2:b":
1162  assumes ‹τ{α1...αn} =df σ{α1...αn}› and ‹σ{τ1...τn}↓› and ‹φ{σ{τ1...τn}}›
1163  shows ‹φ{τ{τ1...τn}}›
1164proof -
1165  AOT_have ‹τ{τ1...τn} = σ{τ1...τn}› using "rule-id-df:1" assms(1,2) by blast
1166  AOT_hence ‹σ{τ1...τn} = τ{τ1...τn}›
1167    using "rule=E" "=I"(1) "t=t-proper:1" "→E" by fast
1168  AOT_thus ‹φ{τ{τ1...τn}}› using assms(3) "rule=E" by blast
1169qed
1170
1171(* TODO: get rid of this, ideally *)
1172AOT_theorem "rule-id-df:2:b[2]":
1173  assumes ‹τ{«(α1,α2)»} =df σ{«(α1,α2)»}› and ‹σ{«(τ1,τ2)»}↓› and ‹φ{σ{«(τ1,τ2)»}}›
1174  shows ‹φ{τ{«(τ1,τ2)»}}›
1175proof -
1176  AOT_have ‹τ{«(τ1,τ2)»} = σ{«(τ1,τ2)»}›
1177  proof -
1178    AOT_have ‹σ{«(τ1,τ2)»}↓ → τ{«(τ1,τ2)»} = σ{«(τ1,τ2)»}›
1179      using assms by (simp add: AOT_sem_conj AOT_sem_imp AOT_sem_eq AOT_sem_not AOT_sem_denotes AOT_model_id_def) (* NOTE: semantics needed *)
1180    AOT_thus ‹τ{«(τ1,τ2)»} = σ{«(τ1,τ2)»}›
1181      using assms(2) "→E" by blast
1182  qed
1183  AOT_hence ‹σ{«(τ1,τ2)»} = τ{«(τ1,τ2)»}›
1184    using "rule=E" "=I"(1) "t=t-proper:1" "→E" by fast
1185  AOT_thus ‹φ{τ{«(τ1,τ2)»}}› using assms(3) "rule=E" by blast
1186qed
1187
1188AOT_theorem "rule-id-df:2:b[zero]":
1189  assumes ‹τ =df σ› and ‹σ↓› and ‹φ{σ}›
1190  shows ‹φ{τ}›
1191proof -
1192  AOT_have ‹τ = σ› using "rule-id-df:1[zero]" assms(1,2) by blast
1193  AOT_hence ‹σ = τ›
1194    using "rule=E" "=I"(1) "t=t-proper:1" "→E" by fast
1195  AOT_thus ‹φ{τ}› using assms(3) "rule=E" by blast
1196qed
1197
1198lemmas "=dfI" = "rule-id-df:2:b" "rule-id-df:2:b[zero]"
1199
1200AOT_theorem "free-thms:1": ‹τ↓ ≡ ∃β (β = τ)›
1201  by (metis "∃E" "rule=I:1" "t=t-proper:2" "→I" "∃I"(1) "≡I" "→E")
1202
1203AOT_theorem "free-thms:2": ‹∀α φ{α} → (∃β (β = τ) → φ{τ})›
1204  by (metis "∃E" "rule=E" "cqt:2[const_var]"[axiom_inst] "→I" "∀E"(1))
1205
1206AOT_theorem "free-thms:3[const_var]": ‹∃β (β = α)›
1207  by (meson "∃I"(2) "id-eq:1")
1208
1209AOT_theorem "free-thms:3[lambda]": assumes ‹INSTANCE_OF_CQT_2(φ)› shows ‹∃β (β = [λν1...νn φ{ν1...νn}])›
1210  by (meson "=I"(3) assms "cqt:2[lambda]"[axiom_inst] "existential:1")
1211
1212AOT_theorem "free-thms:4[rel]": ‹([Π]κ1...κn ∨ κ1...κn[Π]) → ∃β (β = Π)›
1213  by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a"[axiom_inst] "cqt:5:b"[axiom_inst] "→I" "∃I"(1))
1214
1215(* TODO: this is a rather weird way to formulate this and we don't have tuple-existential-elimination
1216         or tuple-equality-elimination in the theory... Splitting them is also a bit unfortunate, though.*)
1217AOT_theorem "free-thms:4[vars]": ‹([Π]κ1...κn ∨ κ1...κn[Π]) → ∃β1...∃βn (β1...βn = κ1...κn)›
1218  by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a"[axiom_inst] "cqt:5:b"[axiom_inst] "→I" "∃I"(1))
1219
1220AOT_theorem "free-thms:4[1,rel]": ‹([Π]κ ∨ κ[Π]) → ∃β (β = Π)›
1221  by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a"[axiom_inst] "cqt:5:b"[axiom_inst] "→I" "∃I"(1))
1222AOT_theorem "free-thms:4[1,1]": ‹([Π]κ ∨ κ[Π]) → ∃β (β = κ)›
1223  by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a"[axiom_inst] "cqt:5:b"[axiom_inst] "→I" "∃I"(1))
1224
1225AOT_theorem "free-thms:4[2,rel]": ‹([Π]κ1κ2 ∨ κ1κ2[Π]) → ∃β (β = Π)›
1226  by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a[2]"[axiom_inst] "cqt:5:b[2]"[axiom_inst] "→I" "∃I"(1))
1227AOT_theorem "free-thms:4[2,1]": ‹([Π]κ1κ2 ∨ κ1κ2[Π]) → ∃β (β = κ1)›
1228  by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[2]"[axiom_inst] "cqt:5:b[2]"[axiom_inst] "→I" "∃I"(1))
1229AOT_theorem "free-thms:4[2,2]": ‹([Π]κ1κ2 ∨ κ1κ2[Π]) → ∃β (β = κ2)›
1230  by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a[2]"[axiom_inst] "cqt:5:b[2]"[axiom_inst] "→I" "∃I"(1))
1231AOT_theorem "free-thms:4[3,rel]": ‹([Π]κ1κ2κ3 ∨ κ1κ2κ3[Π]) → ∃β (β = Π)›
1232  by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a[3]"[axiom_inst] "cqt:5:b[3]"[axiom_inst] "→I" "∃I"(1))
1233AOT_theorem "free-thms:4[3,1]": ‹([Π]κ1κ2κ3 ∨ κ1κ2κ3[Π]) → ∃β (β = κ1)›
1234  by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[3]"[axiom_inst] "cqt:5:b[3]"[axiom_inst] "→I" "∃I"(1))
1235AOT_theorem "free-thms:4[3,2]": ‹([Π]κ1κ2κ3 ∨ κ1κ2κ3[Π]) → ∃β (β = κ2)›
1236  by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[3]"[axiom_inst] "cqt:5:b[3]"[axiom_inst] "→I" "∃I"(1))
1237AOT_theorem "free-thms:4[3,3]": ‹([Π]κ1κ2κ3 ∨ κ1κ2κ3[Π]) → ∃β (β = κ3)›
1238  by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a[3]"[axiom_inst] "cqt:5:b[3]"[axiom_inst] "→I" "∃I"(1))
1239AOT_theorem "free-thms:4[4,rel]": ‹([Π]κ1κ2κ3κ4 ∨ κ1κ2κ3κ4[Π]) → ∃β (β = Π)›
1240  by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a[4]"[axiom_inst] "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
1241AOT_theorem "free-thms:4[4,1]": ‹([Π]κ1κ2κ3κ4 ∨ κ1κ2κ3κ4[Π]) → ∃β (β = κ1)›
1242  by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[4]"[axiom_inst] "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
1243AOT_theorem "free-thms:4[4,2]": ‹([Π]κ1κ2κ3κ4 ∨ κ1κ2κ3κ4[Π]) → ∃β (β = κ2)›
1244  by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[4]"[axiom_inst] "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
1245AOT_theorem "free-thms:4[4,3]": ‹([Π]κ1κ2κ3κ4 ∨ κ1κ2κ3κ4[Π]) → ∃β (β = κ3)›
1246  by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[4]"[axiom_inst] "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
1247AOT_theorem "free-thms:4[4,4]": ‹([Π]κ1κ2κ3κ4 ∨ κ1κ2κ3κ4[Π]) → ∃β (β = κ4)›
1248  by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a[4]"[axiom_inst] "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
1249
1250AOT_theorem "ex:1:a": ‹∀α α↓›
1251  by (rule GEN) (fact "cqt:2[const_var]"[axiom_inst])
1252AOT_theorem "ex:1:b": ‹∀α∃β(β = α)›
1253  by (rule GEN) (fact "free-thms:3[const_var]")
1254
1255AOT_theorem "ex:2:a": ‹□α↓›
1256  by (rule RN) (fact "cqt:2[const_var]"[axiom_inst])
1257AOT_theorem "ex:2:b": ‹□∃β(β = α)›
1258  by (rule RN) (fact "free-thms:3[const_var]")
1259
1260AOT_theorem "ex:3:a": ‹□∀α α↓›
1261  by (rule RN) (fact "ex:1:a")
1262AOT_theorem "ex:3:b": ‹□∀α∃β(β = α)›
1263  by (rule RN) (fact "ex:1:b")
1264
1265AOT_theorem "ex:4:a": ‹∀α □α↓›
1266  by (rule GEN; rule RN) (fact "cqt:2[const_var]"[axiom_inst])
1267AOT_theorem "ex:4:b": ‹∀α□∃β(β = α)›
1268  by (rule GEN; rule RN) (fact "free-thms:3[const_var]")
1269
1270AOT_theorem "ex:5:a": ‹□∀α □α↓›
1271  by (rule RN) (simp add: "ex:4:a")
1272AOT_theorem "ex:5:b": ‹□∀α□∃β(β = α)›
1273  by (rule RN) (simp add: "ex:4:b")
1274
1275AOT_theorem "all-self=:1": ‹□∀α(α = α)›
1276  by (rule RN; rule GEN) (fact "id-eq:1")
1277AOT_theorem "all-self=:2": ‹∀α□(α = α)›
1278  by (rule GEN; rule RN) (fact "id-eq:1")
1279
1280AOT_theorem "id-nec:1": ‹α = β → □(α = β)›
1281proof(rule "→I")
1282  AOT_assume ‹α = β›
1283  moreover AOT_have ‹□(α = α)›
1284    by (rule RN) (fact "id-eq:1")
1285  ultimately AOT_show ‹□(α = β)› using "rule=E" by fast
1286qed
1287
1288AOT_theorem "id-nec:2": ‹τ = σ → □(τ = σ)›
1289proof(rule "→I")
1290  AOT_assume asm: ‹τ = σ›
1291  moreover AOT_have ‹τ↓›
1292    using calculation "t=t-proper:1" "→E" by blast
1293  moreover AOT_have ‹□(τ = τ)›
1294    using calculation "all-self=:2" "∀E"(1) by blast
1295  ultimately AOT_show ‹□(τ = σ)› using "rule=E" by fast
1296qed
1297
1298AOT_theorem "term-out:1": ‹φ{α} ≡ ∃β (β = α & φ{β})›
1299proof (rule "≡I"; rule "→I")
1300  AOT_assume asm: ‹φ{α}›
1301  AOT_show ‹∃β (β = α & φ{β})›
1302    by (rule "∃I"(2)[where β=α]; rule "&I")
1303       (auto simp: "id-eq:1" asm)
1304next
1305  AOT_assume 0: ‹∃β (β = α & φ{β})›
1306  (* TODO: have another look at this instantiation. Ideally AOT_obtain would resolve directly to the existential
1307           statement as proof obligation *)
1308  AOT_obtain β where ‹β = α & φ{β}› using "instantiation"[rotated, OF 0] by blast
1309  AOT_thus ‹φ{α}› using "&E" "rule=E" by blast
1310qed
1311
1312AOT_theorem "term-out:2": ‹τ↓ → (φ{τ} ≡ ∃α(α = τ & φ{α}))›
1313proof(rule "→I")
1314  AOT_assume ‹τ↓›
1315  moreover AOT_have ‹∀α (φ{α} ≡ ∃β (β = α & φ{β}))›
1316    by (rule GEN) (fact "term-out:1")
1317  ultimately AOT_show ‹φ{τ} ≡ ∃α(α = τ & φ{α})›
1318    using "∀E" by blast
1319qed
1320
1321(* TODO: example of an apply-style proof. Keep or reformulate? *)
1322AOT_theorem "term-out:3": ‹(φ{α} & ∀β(φ{β} → β = α)) ≡ ∀β(φ{β} ≡ β = α)›
1323  apply (rule "≡I"; rule "→I")
1324   apply (frule "&E"(1)) apply (drule "&E"(2))
1325   apply (rule GEN; rule "≡I"; rule "→I")
1326  using "rule-ui:2[const_var]" "vdash-properties:5" apply blast
1327  apply (meson "rule=E" "id-eq:1")
1328  apply (rule "&I")
1329  using "id-eq:1" "≡E"(2) "rule-ui:3" apply blast
1330  apply (rule GEN; rule "→I")
1331  using "≡E"(1) "rule-ui:2[const_var]" by blast
1332
1333AOT_theorem "term-out:4": ‹(φ{β} & ∀α(φ{α} → α = β)) ≡ ∀α(φ{α} ≡ α = β)›
1334  using "term-out:3" . (* TODO: same as above - another instance of the generalized alphabetic variant rule... *)
1335
1336(* TODO: would of course be nice to define it without the syntax magic *)
1337AOT_define AOT_exists_unique :: ‹α ⇒ φ ⇒ φ›
1338  "uniqueness:1": ‹«AOT_exists_unique φ» ≡df ∃α (φ{α} & ∀β (φ{β} → β = α))›
1339syntax "_AOT_exists_unique" :: ‹α ⇒ φ ⇒ φ› ("∃!_ _" [1,40])
1340AOT_syntax_print_translations
1341  "_AOT_exists_unique τ φ" <= "CONST AOT_exists_unique (_abs τ φ)"
1342syntax
1343   "_AOT_exists_unique_ellipse" :: ‹id_position ⇒ id_position ⇒ φ ⇒ φ› (‹∃!_...∃!_ _› [1,40])
1344parse_ast_translation‹[(syntax_const‹_AOT_exists_unique_ellipse›, fn ctx => fn [a,b,c] =>
1345  Ast.mk_appl (Ast.Constant "AOT_exists_unique") [parseEllipseList "_AOT_vars" ctx [a,b],c]),
1346(syntax_const‹_AOT_exists_unique›,AOT_restricted_binder const_name‹AOT_exists_unique› const_syntax‹AOT_conj›)]›
1347print_translation‹AOT_syntax_print_translations
1348  [AOT_preserve_binder_abs_tr' const_syntax‹AOT_exists_unique› syntax_const‹_AOT_exists_unique› (syntax_const‹_AOT_exists_unique_ellipse›, true) const_name‹AOT_conj›,
1349  AOT_binder_trans @{theory} @{binding "AOT_exists_unique_binder"} syntax_const‹_AOT_exists_unique›]
1350›
1351
1352
1353context AOT_meta_syntax
1354begin
1355notation AOT_exists_unique (binder "∃!" 20)
1356end
1357context AOT_no_meta_syntax
1358begin
1359no_notation AOT_exists_unique (binder "∃!" 20)
1360end
1361
1362AOT_theorem "uniqueness:2": ‹∃!α φ{α} ≡ ∃α∀β(φ{β} ≡ β = α)›
1363proof(rule "≡I"; rule "→I")
1364    AOT_assume ‹∃!α φ{α}›
1365    AOT_hence ‹∃α (φ{α} & ∀β (φ{β} → β = α))›
1366      using "uniqueness:1" "≡dfE" by blast
1367    then AOT_obtain α where ‹φ{α} & ∀β (φ{β} → β = α)› using "instantiation"[rotated] by blast
1368    AOT_hence ‹∀β(φ{β} ≡ β = α)› using "term-out:3" "≡E" by blast
1369    AOT_thus ‹∃α∀β(φ{β} ≡ β = α)›
1370      using "∃I" by fast
1371next
1372    AOT_assume ‹∃α∀β(φ{β} ≡ β = α)›
1373    then AOT_obtain α where ‹∀β (φ{β} ≡ β = α)› using "instantiation"[rotated] by blast
1374    AOT_hence ‹φ{α} & ∀β (φ{β} → β = α)› using "term-out:3" "≡E" by blast
1375    AOT_hence ‹∃α (φ{α} & ∀β (φ{β} → β = α))›
1376      using "∃I" by fast
1377    AOT_thus ‹∃!α φ{α}›
1378      using "uniqueness:1" "≡dfI" by blast
1379qed
1380
1381AOT_theorem "uni-most": ‹∃!α φ{α} → ∀β∀γ((φ{β} & φ{γ}) → β = γ)›
1382proof(rule "→I"; rule GEN; rule GEN; rule "→I")
1383  fix β γ
1384  AOT_assume ‹∃!α φ{α}›
1385  AOT_hence ‹∃α∀β(φ{β} ≡ β = α)›
1386    using "uniqueness:2" "≡E" by blast
1387  then AOT_obtain α where ‹∀β(φ{β} ≡ β = α)›
1388    using "instantiation"[rotated] by blast
1389  moreover AOT_assume ‹φ{β} & φ{γ}›
1390  ultimately AOT_have ‹β = α› and ‹γ = α›
1391    using "∀E"(2) "&E" "≡E"(1,2) by blast+
1392  AOT_thus ‹β = γ›
1393    by (metis "rule=E" "id-eq:2" "→E")
1394qed
1395
1396AOT_theorem "nec-exist-!": ‹∀α(φ{α} → □φ{α}) → (∃!α φ{α} → ∃!α □φ{α})›
1397proof (rule "→I"; rule "→I")
1398  AOT_assume a: ‹∀α(φ{α} → □φ{α})›
1399  AOT_assume ‹∃!α φ{α}›
1400  AOT_hence ‹∃α (φ{α} & ∀β (φ{β} → β = α))› using "uniqueness:1" "≡dfE" by blast
1401  then AOT_obtain α where ξ: ‹φ{α} & ∀β (φ{β} → β = α)› using "instantiation"[rotated] by blast
1402  AOT_have ‹□φ{α}›
1403    using ξ a "&E" "∀E" "→E" by fast
1404  moreover AOT_have ‹∀β (□φ{β} → β = α)›
1405    apply (rule GEN; rule "→I")
1406    using ξ[THEN "&E"(2), THEN "∀E"(2), THEN "→E"] "qml:2"[axiom_inst, THEN "→E"] by blast
1407  ultimately AOT_have ‹(□φ{α} & ∀β (□φ{β} → β = α))›
1408    using "&I" by blast
1409  AOT_thus ‹∃!α □φ{α}›
1410    using "uniqueness:1" "≡dfI" "∃I" by fast
1411qed
1412
1413AOT_theorem "act-cond": ‹𝒜(φ → ψ) → (𝒜φ → 𝒜ψ)›
1414  using "→I" "≡E"(1) "logic-actual-nec:2"[axiom_inst] by blast
1415
1416AOT_theorem "nec-imp-act": ‹□φ → 𝒜φ›
1417  by (metis "act-cond" "contraposition:1[2]" "≡E"(4) "qml:2"[THEN act_closure, axiom_inst] "qml-act:2"[axiom_inst] RAA(1) "→E" "→I")
1418
1419AOT_theorem "act-conj-act:1": ‹𝒜(𝒜φ → φ)›
1420  using "→I" "≡E"(2) "logic-actual-nec:2"[axiom_inst] "logic-actual-nec:4"[axiom_inst] by blast
1421
1422AOT_theorem "act-conj-act:2": ‹𝒜(φ → 𝒜φ)›
1423  by (metis "→I" "≡E"(2, 4) "logic-actual-nec:2"[axiom_inst] "logic-actual-nec:4"[axiom_inst] RAA(1))
1424
1425AOT_theorem "act-conj-act:3": ‹(𝒜φ & 𝒜ψ) → 𝒜(φ & ψ)›
1426proof -
1427  AOT_have ‹□(φ → (ψ → (φ & ψ)))›
1428    by (rule RN) (fact Adjunction)
1429  AOT_hence ‹𝒜(φ → (ψ → (φ & ψ)))›
1430    using "nec-imp-act" "→E" by blast
1431  AOT_hence ‹𝒜φ → 𝒜(ψ → (φ & ψ))›
1432    using "act-cond" "→E" by blast
1433  moreover AOT_have ‹𝒜(ψ → (φ & ψ)) → (𝒜ψ → 𝒜(φ & ψ))›
1434    by (fact "act-cond")
1435  ultimately AOT_have ‹𝒜φ → (𝒜ψ → 𝒜(φ & ψ))›
1436    using "→I" "→E" by metis
1437  AOT_thus ‹(𝒜φ & 𝒜ψ) → 𝒜(φ & ψ)›
1438    by (metis Importation "→E")
1439qed
1440
1441AOT_theorem "act-conj-act:4": ‹𝒜(𝒜φ ≡ φ)›
1442proof -
1443  AOT_have ‹(𝒜(𝒜φ → φ) & 𝒜(φ → 𝒜φ)) → 𝒜((𝒜φ → φ) & (φ → 𝒜φ))›
1444    by (fact "act-conj-act:3")
1445  moreover AOT_have ‹𝒜(𝒜φ → φ) & 𝒜(φ → 𝒜φ)›
1446    using "&I" "act-conj-act:1" "act-conj-act:2" by simp
1447  ultimately AOT_have ζ: ‹𝒜((𝒜φ → φ) & (φ → 𝒜φ))›
1448    using "→E" by blast
1449  AOT_have ‹𝒜(((𝒜φ → φ) & (φ → 𝒜φ)) → (𝒜φ ≡ φ))›
1450    using "conventions:3"[THEN "df-rules-formulas[2]", THEN act_closure, axiom_inst] by blast
1451  AOT_hence ‹𝒜((𝒜φ → φ) & (φ → 𝒜φ)) → 𝒜(𝒜φ ≡ φ)›
1452    using "act-cond" "→E" by blast
1453  AOT_thus ‹𝒜(𝒜φ ≡ φ)› using ζ "→E" by blast
1454qed
1455
1456(* TODO: consider introducing AOT_inductive *)
1457inductive arbitrary_actualization for φ where
1458  ‹arbitrary_actualization φ «𝒜φ»›
1459| ‹arbitrary_actualization φ «𝒜ψ»› if ‹arbitrary_actualization φ ψ›
1460declare arbitrary_actualization.cases[AOT] arbitrary_actualization.induct[AOT]
1461        arbitrary_actualization.simps[AOT] arbitrary_actualization.intros[AOT]
1462syntax arbitrary_actualization :: ‹φ' ⇒ φ' ⇒ AOT_prop› ("ARBITRARY'_ACTUALIZATION'(_,_')")
1463
1464notepad
1465begin
1466  AOT_modally_strict {
1467    fix φ
1468    AOT_have ‹ARBITRARY_ACTUALIZATION(𝒜φ ≡ φ, 𝒜(𝒜φ ≡ φ))›
1469      using AOT_PLM.arbitrary_actualization.intros by metis
1470    AOT_have ‹ARBITRARY_ACTUALIZATION(𝒜φ ≡ φ, 𝒜𝒜(𝒜φ ≡ φ))›
1471      using AOT_PLM.arbitrary_actualization.intros by metis
1472    AOT_have ‹ARBITRARY_ACTUALIZATION(𝒜φ ≡ φ, 𝒜𝒜𝒜(𝒜φ ≡ φ))›
1473      using AOT_PLM.arbitrary_actualization.intros by metis
1474  }
1475end
1476
1477
1478AOT_theorem "closure-act:1": assumes ‹ARBITRARY_ACTUALIZATION(𝒜φ ≡ φ, ψ)› shows ‹ψ›
1479using assms proof(induct)
1480  case 1
1481  AOT_show ‹𝒜(𝒜φ ≡ φ)›
1482    by (simp add: "act-conj-act:4")
1483next
1484  case (2 ψ)
1485  AOT_thus ‹𝒜ψ›
1486    by (metis arbitrary_actualization.simps "≡E"(1) "logic-actual-nec:4"[axiom_inst])
1487qed
1488
1489AOT_theorem "closure-act:2": ‹∀α 𝒜(𝒜φ{α} ≡ φ{α})›
1490  by (simp add: "act-conj-act:4" "∀I")
1491
1492AOT_theorem "closure-act:3": ‹𝒜∀α 𝒜(𝒜φ{α} ≡ φ{α})›
1493  by (metis (no_types, lifting) "act-conj-act:4" "≡E"(1,2) "logic-actual-nec:3"[axiom_inst] "logic-actual-nec:4"[axiom_inst] "∀I")
1494
1495AOT_theorem "closure-act:4": ‹𝒜∀α1...∀αn 𝒜(𝒜φ{α1...αn} ≡ φ{α1...αn})›
1496  using "closure-act:3" .
1497
1498(* TODO: examine these proofs *)
1499AOT_theorem "RA[1]": assumes ‹⊢ φ› shows ‹⊢ 𝒜φ›
1500  (* This proof is the one rejected in remark (136) (meta-rule) *)
1501  using "¬¬E" assms "≡E"(3) "logic-actual"[act_axiom_inst] "logic-actual-nec:1"[axiom_inst] "modus-tollens:2" by blast
1502AOT_theorem "RA[2]": assumes ‹⊢□ φ› shows ‹𝒜φ›
1503  (* This is actually Γ ⊢□ φ ⟹ □Γ ⊢□ 𝒜φ*)
1504  using RN assms "nec-imp-act" "vdash-properties:5" by blast
1505AOT_theorem "RA[3]": assumes ‹Γ ⊢□ φ› shows ‹𝒜Γ ⊢□ 𝒜φ›
1506  using assms by (meson AOT_sem_act imageI)
1507  (* This is not exactly right either. *)
1508
1509AOT_act_theorem "ANeg:1": ‹¬𝒜φ ≡ ¬φ›
1510  by (simp add: "RA[1]" "contraposition:1[1]" "deduction-theorem" "≡I" "logic-actual"[act_axiom_inst])
1511
1512AOT_act_theorem "ANeg:2": ‹¬𝒜¬φ ≡ φ›
1513  using "ANeg:1" "≡I" "≡E"(5) "useful-tautologies:1" "useful-tautologies:2" by blast
1514
1515AOT_theorem "Act-Basic:1": ‹𝒜φ ∨ 𝒜¬φ›
1516  by (meson "∨I"(1,2) "≡E"(2) "logic-actual-nec:1"[axiom_inst] "raa-cor:1")
1517
1518AOT_theorem "Act-Basic:2": ‹𝒜(φ & ψ) ≡ (𝒜φ & 𝒜ψ)›
1519proof (rule "≡I"; rule "→I")
1520  AOT_assume ‹𝒜(φ & ψ)›
1521  moreover AOT_have ‹𝒜((φ & ψ) → φ)›
1522    by (simp add: "RA[2]" "Conjunction Simplification"(1))
1523  moreover AOT_have ‹𝒜((φ & ψ) → ψ)›
1524    by (simp add: "RA[2]" "Conjunction Simplification"(2))
1525  ultimately AOT_show ‹𝒜φ & 𝒜ψ›
1526    using "act-cond"[THEN "→E", THEN "→E"] "&I" by metis
1527next
1528  AOT_assume ‹𝒜φ & 𝒜ψ›
1529  AOT_thus ‹𝒜(φ & ψ)›
1530    using "act-conj-act:3" "vdash-properties:6" by blast
1531qed
1532
1533AOT_theorem "Act-Basic:3": ‹𝒜(φ ≡ ψ) ≡ (𝒜(φ → ψ) & 𝒜(ψ → φ))›
1534proof (rule "≡I"; rule "→I")
1535  AOT_assume ‹𝒜(φ ≡ ψ)›
1536  moreover AOT_have ‹𝒜((φ ≡ ψ) → (φ → ψ))›
1537    by (simp add: "RA[2]" "deduction-theorem" "≡E"(1))
1538  moreover AOT_have ‹𝒜((φ ≡ ψ) → (ψ → φ))›
1539    by (simp add: "RA[2]" "deduction-theorem" "≡E"(2))
1540  ultimately AOT_show ‹𝒜(φ → ψ) & 𝒜(ψ → φ)›
1541    using "act-cond"[THEN "→E", THEN "→E"] "&I" by metis
1542next
1543  AOT_assume ‹𝒜(φ → ψ) & 𝒜(ψ → φ)›
1544  AOT_hence ‹𝒜((φ → ψ) & (ψ → φ))›
1545    by (metis "act-conj-act:3" "vdash-properties:10")
1546  moreover AOT_have ‹𝒜(((φ → ψ) & (ψ → φ)) → (φ ≡ ψ))›
1547    by (simp add: "conventions:3" "RA[2]" "df-rules-formulas[2]" "vdash-properties:1[2]")
1548  ultimately AOT_show ‹𝒜(φ ≡ ψ)›
1549    using "act-cond"[THEN "→E", THEN "→E"] by metis
1550qed
1551
1552AOT_theorem "Act-Basic:4": ‹(𝒜(φ → ψ) & 𝒜(ψ → φ)) ≡ (𝒜φ ≡ 𝒜ψ)›
1553proof (rule "≡I"; rule "→I")
1554  AOT_assume 0: ‹𝒜(φ → ψ) & 𝒜(ψ → φ)›
1555  AOT_show ‹𝒜φ ≡ 𝒜ψ›
1556    using 0 "&E" "act-cond"[THEN "→E", THEN "→E"] "≡I" "→I" by metis
1557next
1558  AOT_assume ‹𝒜φ ≡ 𝒜ψ›
1559  AOT_thus ‹𝒜(φ → ψ) & 𝒜(ψ → φ)›
1560    by (metis "→I" "logic-actual-nec:2"[axiom_inst] "≡E"(1,2) "&I")
1561qed
1562
1563AOT_theorem "Act-Basic:5": ‹𝒜(φ ≡ ψ) ≡ (𝒜φ ≡ 𝒜ψ)›
1564  using "Act-Basic:3" "Act-Basic:4" "≡E"(5) by blast
1565
1566AOT_theorem "Act-Basic:6": ‹𝒜φ ≡ □𝒜φ›
1567  by (simp add: "≡I" "qml:2"[axiom_inst] "qml-act:1"[axiom_inst])
1568
1569AOT_theorem "Act-Basic:7": ‹𝒜□φ → □𝒜φ›
1570  by (metis "Act-Basic:6" "→I" "→E" "≡E"(1,2) "nec-imp-act" "qml-act:2"[axiom_inst])
1571
1572AOT_theorem "Act-Basic:8": ‹□φ → □𝒜φ›
1573  using "Hypothetical Syllogism" "nec-imp-act" "qml-act:1"[axiom_inst] by blast
1574
1575AOT_theorem "Act-Basic:9": ‹𝒜(φ ∨ ψ) ≡ (𝒜φ ∨ 𝒜ψ)›
1576proof (rule "≡I"; rule "→I")
1577  AOT_assume ‹𝒜(φ ∨ ψ)›
1578  AOT_thus ‹𝒜φ ∨ 𝒜ψ›
1579  proof (rule "raa-cor:3")
1580    AOT_assume ‹¬(𝒜φ ∨ 𝒜ψ)›
1581    AOT_hence ‹¬𝒜φ & ¬𝒜ψ›
1582      by (metis "≡E"(1) "oth-class-taut:5:d")
1583    AOT_hence ‹𝒜¬φ & 𝒜¬ψ›
1584      using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] "&E" "&I" by metis
1585    AOT_hence ‹𝒜(¬φ & ¬ψ)›
1586      using "≡E" "Act-Basic:2" by metis
1587    moreover AOT_have ‹𝒜((¬φ & ¬ψ) ≡ ¬(φ ∨ ψ))›
1588      using "RA[2]" "≡E"(6) "oth-class-taut:3:a" "oth-class-taut:5:d" by blast
1589    moreover AOT_have ‹𝒜(¬φ & ¬ψ) ≡ 𝒜(¬(φ ∨ ψ))›
1590      using calculation(2) by (metis "Act-Basic:5" "≡E"(1))
1591    ultimately AOT_have ‹𝒜(¬(φ ∨ ψ))› using "≡E" by blast
1592    AOT_thus ‹¬𝒜(φ ∨ ψ)›
1593      using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(1)] by auto
1594  qed
1595next
1596  AOT_assume ‹𝒜φ ∨ 𝒜ψ›
1597  AOT_thus ‹𝒜(φ ∨ ψ)›
1598    by (meson "RA[2]" "act-cond" "∨I"(1) "∨E"(1) "Disjunction Addition"(1) "Disjunction Addition"(2))
1599qed
1600
1601AOT_theorem "Act-Basic:10": ‹𝒜∃α φ{α} ≡ ∃α 𝒜φ{α}›
1602proof -
1603  AOT_have θ: ‹¬𝒜∀α ¬φ{α} ≡ ¬∀α 𝒜¬φ{α}›
1604    by (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
1605       (metis "logic-actual-nec:3"[axiom_inst])
1606  AOT_have ξ: ‹¬∀α 𝒜¬φ{α} ≡ ¬∀α ¬𝒜φ{α}›
1607    by (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
1608       (rule "logic-actual-nec:1"[THEN universal_closure, axiom_inst, THEN "cqt-basic:3"[THEN "→E"]])
1609  AOT_have ‹𝒜(∃α φ{α}) ≡ 𝒜(¬∀α ¬φ{α})›
1610    using "conventions:4"[THEN "df-rules-formulas[1]", THEN act_closure, axiom_inst]
1611          "conventions:4"[THEN "df-rules-formulas[2]", THEN act_closure, axiom_inst]
1612    "Act-Basic:4"[THEN "≡E"(1)] "&I" "Act-Basic:5"[THEN "≡E"(2)] by metis
1613  also AOT_have ‹… ≡ ¬𝒜∀α ¬φ{α}›
1614    by (simp add: "logic-actual-nec:1" "vdash-properties:1[2]")
1615  also AOT_have ‹… ≡ ¬∀α 𝒜 ¬φ{α}› using θ by blast
1616  also AOT_have ‹… ≡ ¬∀α ¬𝒜 φ{α}› using ξ by blast
1617  also AOT_have ‹… ≡ ∃α 𝒜 φ{α}›
1618    using "conventions:4"[THEN "≡Df"] by (metis "≡E"(6) "oth-class-taut:3:a")
1619  finally AOT_show ‹𝒜∃α φ{α} ≡ ∃α 𝒜φ{α}› .
1620qed
1621
1622
1623AOT_theorem "Act-Basic:11": ‹𝒜∀α(φ{α} ≡ ψ{α}) ≡ ∀α(𝒜φ{α} ≡ 𝒜ψ{α})›
1624proof(rule "≡I"; rule "→I")
1625  AOT_assume ‹𝒜∀α(φ{α} ≡ ψ{α})›
1626  AOT_hence ‹∀α𝒜(φ{α} ≡ ψ{α})›
1627    using "logic-actual-nec:3"[axiom_inst, THEN "≡E"(1)] by blast
1628  AOT_hence ‹𝒜(φ{α} ≡ ψ{α})› for α using "∀E" by blast
1629  AOT_hence ‹𝒜φ{α} ≡ 𝒜ψ{α}› for α by (metis "Act-Basic:5" "≡E"(1))
1630  AOT_thus ‹∀α(𝒜φ{α} ≡ 𝒜ψ{α})› by (rule "∀I")
1631next
1632  AOT_assume ‹∀α(𝒜φ{α} ≡ 𝒜ψ{α})›
1633  AOT_hence ‹𝒜φ{α} ≡ 𝒜ψ{α}› for α using "∀E" by blast
1634  AOT_hence ‹𝒜(φ{α} ≡ ψ{α})› for α by (metis "Act-Basic:5" "≡E"(2))
1635  AOT_hence ‹∀α 𝒜(φ{α} ≡ ψ{α})› by (rule "∀I")
1636  AOT_thus ‹𝒜∀α(φ{α} ≡ ψ{α})›
1637    using "logic-actual-nec:3"[axiom_inst, THEN "≡E"(2)] by fast
1638qed
1639
1640AOT_act_theorem "act-quant-uniq": ‹∀β(𝒜φ{β} ≡ β = α) ≡ ∀β(φ{β} ≡ β = α)›
1641proof(rule "≡I"; rule "→I")
1642  AOT_assume ‹∀β(𝒜φ{β} ≡ β = α)›
1643  AOT_hence ‹𝒜φ{β} ≡ β = α› for β using "∀E" by blast
1644  AOT_hence ‹φ{β} ≡ β = α› for β
1645    using "≡I" "→I" "RA[1]" "≡E"(1) "≡E"(2) "logic-actual"[act_axiom_inst] "vdash-properties:6"
1646    by metis
1647  AOT_thus ‹∀β(φ{β} ≡ β = α)› by (rule "∀I")
1648next
1649  AOT_assume ‹∀β(φ{β} ≡ β = α)›
1650  AOT_hence ‹φ{β} ≡ β = α› for β using "∀E" by blast
1651  AOT_hence ‹𝒜φ{β} ≡ β = α› for β
1652    using "≡I" "→I" "RA[1]" "≡E"(1) "≡E"(2) "logic-actual"[act_axiom_inst] "vdash-properties:6"
1653    by metis
1654  AOT_thus ‹∀β(𝒜φ{β} ≡ β = α)› by (rule "∀I")
1655qed
1656
1657AOT_act_theorem "fund-cont-desc": ‹x = ιx(φ{x}) ≡ ∀z(φ{z} ≡ z = x)›
1658  using descriptions[axiom_inst] "act-quant-uniq" "≡E"(5) by fast
1659
1660AOT_act_theorem hintikka: ‹x = ιx(φ{x}) ≡ (φ{x} & ∀z (φ{z} → z = x))›
1661  using "Commutativity of ≡"[THEN "≡E"(1)] "term-out:3" "fund-cont-desc" "≡E"(5) by blast
1662
1663
1664locale russel_axiom =
1665  fixes ψ
1666  assumes ψ_denotes_asm: "[v ⊨ ψ{κ}] ⟹ [v ⊨ κ↓]"
1667begin
1668AOT_act_theorem "russell-axiom": ‹ψ{ιx φ{x}} ≡ ∃x(φ{x} & ∀z(φ{z} → z = x) & ψ{x})›
1669proof -
1670  AOT_have b: ‹∀x (x = ιx φ{x} ≡ (φ{x} & ∀z(φ{z} → z = x)))›
1671    using hintikka "∀I" by fast
1672  show ?thesis
1673  proof(rule "≡I"; rule "→I")
1674    AOT_assume c: ‹ψ{ιx φ{x}}›
1675    AOT_hence d: ‹ιx φ{x}↓› using ψ_denotes_asm by blast
1676    AOT_hence ‹∃y (y = ιx φ{x})› by (metis "rule=I:1" "existential:1")
1677    then AOT_obtain a where a_def: ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1678    moreover AOT_have ‹a = ιx φ{x} ≡ (φ{a} & ∀z(φ{z} → z = a))› using b "∀E" by blast
1679    ultimately AOT_have ‹φ{a} & ∀z(φ{z} → z = a)› using "≡E" by blast
1680    moreover AOT_have ‹ψ{a}›
1681    proof - 
1682      AOT_have 1: ‹∀x∀y(x = y → y = x)›
1683        by (simp add: "id-eq:2" "universal-cor")
1684      AOT_have ‹a = ιx φ{x} →  ιx φ{x} = a›
1685        by (rule "∀E"(1)[where τ="«ιx φ{x}»"]; rule "∀E"(2)[where β=a])
1686           (auto simp: 1 d "universal-cor")
1687      AOT_thus ‹ψ{a}›
1688        using a_def c "rule=E" "→E" by blast
1689    qed
1690    ultimately AOT_have ‹φ{a} & ∀z(φ{z} → z = a) & ψ{a}› by (rule "&I")
1691    AOT_thus ‹∃x(φ{x} & ∀z(φ{z} → z = x) & ψ{x})› by (rule "∃I")
1692  next
1693    AOT_assume ‹∃x(φ{x} & ∀z(φ{z} → z = x) & ψ{x})›
1694    then AOT_obtain b where g: ‹φ{b} & ∀z(φ{z} → z = b) & ψ{b}› using "instantiation"[rotated] by blast
1695    AOT_hence h: ‹b = ιx φ{x} ≡ (φ{b} & ∀z(φ{z} → z = b))› using b "∀E" by blast
1696    AOT_have ‹φ{b} & ∀z(φ{z} → z = b)› and j: ‹ψ{b}› using g "&E" by blast+
1697    AOT_hence ‹b = ιx φ{x}› using h "≡E" by blast
1698    AOT_thus ‹ψ{ιx φ{x}}› using j "rule=E" by blast
1699  qed
1700qed
1701end
1702
1703(* TODO: this nicely shows off using locales with the embedding, but maybe there is still a nicer way *)
1704(* TODO: sledgehammer tends to refer to ψ_denotes_asm in these instantiation instead of referring
1705         to cqt:5:a - should be fixed *)
1706interpretation "russell-axiom[exe,1]": russel_axiom ‹λ κ . «[Π]κ»›
1707  by standard (metis "cqt:5:a[1]"[axiom_inst, THEN "→E"] "&E"(2))
1708interpretation "russell-axiom[exe,2,1,1]": russel_axiom ‹λ κ . «[Π]κκ'»›
1709  by standard (metis "cqt:5:a[2]"[axiom_inst, THEN "→E"] "&E")
1710interpretation "russell-axiom[exe,2,1,2]": russel_axiom ‹λ κ . «[Π]κ'κ»›
1711  by standard (metis "cqt:5:a[2]"[axiom_inst, THEN "→E"] "&E"(2))
1712interpretation "russell-axiom[exe,2,2]": russel_axiom ‹λ κ . «[Π]κκ»›
1713  by standard (metis "cqt:5:a[2]"[axiom_inst, THEN "→E"] "&E"(2))
1714interpretation "russell-axiom[exe,3,1,1]": russel_axiom ‹λ κ . «[Π]κκ'κ''»›
1715  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E")
1716interpretation "russell-axiom[exe,3,1,2]": russel_axiom ‹λ κ . «[Π]κ'κκ''»›
1717  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E")
1718interpretation "russell-axiom[exe,3,1,3]": russel_axiom ‹λ κ . «[Π]κ'κ''κ»›
1719  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E"(2))
1720interpretation "russell-axiom[exe,3,2,1]": russel_axiom ‹λ κ . «[Π]κκκ'»›
1721  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E")
1722interpretation "russell-axiom[exe,3,2,2]": russel_axiom ‹λ κ . «[Π]κκ'κ»›
1723  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E"(2))
1724interpretation "russell-axiom[exe,3,2,3]": russel_axiom ‹λ κ . «[Π]κ'κκ»›
1725  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E"(2))
1726interpretation "russell-axiom[exe,3,3]": russel_axiom ‹λ κ . «[Π]κκκ»›
1727  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E"(2))
1728
1729interpretation "russell-axiom[enc,1]": russel_axiom ‹λ κ . «κ[Π]»›
1730  by standard (metis "cqt:5:b[1]"[axiom_inst, THEN "→E"] "&E"(2))
1731interpretation "russell-axiom[enc,2,1]": russel_axiom ‹λ κ . «κκ'[Π]»›
1732  by standard (metis "cqt:5:b[2]"[axiom_inst, THEN "→E"] "&E")
1733interpretation "russell-axiom[enc,2,2]": russel_axiom ‹λ κ . «κ'κ[Π]»›
1734  by standard (metis "cqt:5:b[2]"[axiom_inst, THEN "→E"] "&E"(2))
1735interpretation "russell-axiom[enc,2,3]": russel_axiom ‹λ κ . «κκ[Π]»›
1736  by standard (metis "cqt:5:b[2]"[axiom_inst, THEN "→E"] "&E"(2))
1737interpretation "russell-axiom[enc,3,1,1]": russel_axiom ‹λ κ . «κκ'κ''[Π]»›
1738  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E")
1739interpretation "russell-axiom[enc,3,1,2]": russel_axiom ‹λ κ . «κ'κκ''[Π]»›
1740  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E")
1741interpretation "russell-axiom[enc,3,1,3]": russel_axiom ‹λ κ . «κ'κ''κ[Π]»›
1742  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E"(2))
1743interpretation "russell-axiom[enc,3,2,1]": russel_axiom ‹λ κ . «κκκ'[Π]»›
1744  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E")
1745interpretation "russell-axiom[enc,3,2,2]": russel_axiom ‹λ κ . «κκ'κ[Π]»›
1746  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E"(2))
1747interpretation "russell-axiom[enc,3,2,3]": russel_axiom ‹λ κ . «κ'κκ[Π]»›
1748  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E"(2))
1749interpretation "russell-axiom[enc,3,3]": russel_axiom ‹λ κ . «κκκ[Π]»›
1750  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E"(2))
1751
1752AOT_act_theorem "1-exists:1": ‹ιx φ{x}↓ ≡ ∃!x φ{x}›
1753proof(rule "≡I"; rule "→I")
1754  AOT_assume ‹ιx φ{x}↓›
1755  AOT_hence ‹∃y (y = ιx φ{x})› by (metis "rule=I:1" "existential:1")
1756  then AOT_obtain a where ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1757  AOT_hence ‹φ{a} & ∀z (φ{z} → z = a)› using hintikka "≡E" by blast
1758  AOT_hence ‹∃x (φ{x} & ∀z (φ{z} → z = x))› by (rule "∃I")
1759  AOT_thus ‹∃!x φ{x}› using "uniqueness:1"[THEN "≡dfI"] by blast
1760next
1761  AOT_assume ‹∃!x φ{x}›
1762  AOT_hence ‹∃x (φ{x} & ∀z (φ{z} → z = x))›
1763    using "uniqueness:1"[THEN "≡dfE"] by blast
1764  then AOT_obtain b where ‹φ{b} & ∀z (φ{z} → z = b)› using "instantiation"[rotated] by blast
1765  AOT_hence ‹b = ιx φ{x}› using hintikka "≡E" by blast
1766  AOT_thus ‹ιx φ{x}↓› by (metis "t=t-proper:2" "vdash-properties:6")
1767qed
1768
1769AOT_act_theorem "1-exists:2": ‹∃y(y=ιx φ{x}) ≡ ∃!x φ{x}›
1770  using "1-exists:1" "free-thms:1" "≡E"(6) by blast
1771
1772AOT_act_theorem "y-in:1": ‹x = ιx φ{x} → φ{x}›
1773  using "&E"(1) "→I" hintikka "≡E"(1) by blast
1774
1775AOT_act_theorem "y-in:2": ‹z = ιx φ{x} → φ{z}› using "y-in:1". (* TODO: same as above *)
1776
1777AOT_act_theorem "y-in:3": ‹ιx φ{x}↓ → φ{ιx φ{x}}›
1778proof(rule "→I")
1779  AOT_assume ‹ιx φ{x}↓›
1780  AOT_hence ‹∃y (y = ιx φ{x})› by (metis "rule=I:1" "existential:1")
1781  then AOT_obtain a where ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1782  moreover AOT_have ‹φ{a}› using calculation hintikka "≡E"(1) "&E" by blast
1783  ultimately AOT_show ‹φ{ιx φ{x}}› using "rule=E" by blast
1784qed
1785
1786AOT_act_theorem "y-in:4": ‹∃y (y = ιx φ{x}) → φ{ιx φ{x}}›
1787  using "y-in:3"[THEN "→E"] "free-thms:1"[THEN "≡E"(2)] "→I" by blast
1788
1789
1790AOT_theorem "act-quant-nec": ‹∀β (𝒜φ{β} ≡ β = α) ≡ ∀β(𝒜𝒜φ{β} ≡ β = α)›
1791proof(rule "≡I"; rule "→I")
1792  AOT_assume ‹∀β (𝒜φ{β} ≡ β = α)›
1793  AOT_hence ‹𝒜φ{β} ≡ β = α› for β using "∀E" by blast
1794  AOT_hence ‹𝒜𝒜φ{β} ≡ β = α› for β 
1795    by (metis "Act-Basic:5" "act-conj-act:4" "≡E"(1) "≡E"(5))
1796  AOT_thus ‹∀β(𝒜𝒜φ{β} ≡ β = α)›
1797    by (rule "∀I")
1798next
1799  AOT_assume ‹∀β(𝒜𝒜φ{β} ≡ β = α)›
1800  AOT_hence ‹𝒜𝒜φ{β} ≡ β = α› for β using "∀E" by blast
1801  AOT_hence ‹𝒜φ{β} ≡ β = α› for β
1802    by (metis "Act-Basic:5" "act-conj-act:4" "≡E"(1) "≡E"(6))
1803  AOT_thus ‹∀β (𝒜φ{β} ≡ β = α)›
1804    by (rule "∀I")
1805qed
1806
1807AOT_theorem "equi-desc-descA:1": ‹x = ιx φ{x} ≡ x = ιx(𝒜φ{x})›
1808proof -
1809  AOT_have ‹x = ιx φ{x} ≡ ∀z (𝒜φ{z} ≡ z = x)›  using descriptions[axiom_inst] by blast
1810  also AOT_have ‹... ≡ ∀z (𝒜𝒜φ{z} ≡ z = x)›
1811  proof(rule "≡I"; rule "→I"; rule "∀I")
1812    AOT_assume ‹∀z (𝒜φ{z} ≡ z = x)›
1813    AOT_hence ‹𝒜φ{a} ≡ a = x› for a using "∀E" by blast
1814    AOT_thus ‹𝒜𝒜φ{a} ≡ a = x› for a by (metis "Act-Basic:5" "act-conj-act:4" "≡E"(1) "≡E"(5))
1815  next
1816    AOT_assume ‹∀z (𝒜𝒜φ{z} ≡ z = x)›
1817    AOT_hence ‹𝒜𝒜φ{a} ≡ a = x› for a using "∀E" by blast
1818    AOT_thus ‹𝒜φ{a} ≡ a = x› for a  by (metis "Act-Basic:5" "act-conj-act:4" "≡E"(1) "≡E"(6))
1819  qed
1820  also AOT_have ‹... ≡ x = ιx(𝒜φ{x})›
1821    using "Commutativity of ≡"[THEN "≡E"(1)] descriptions[axiom_inst] by fast
1822  finally show ?thesis .
1823qed
1824
1825AOT_theorem "equi-desc-descA:2": ‹ιx φ{x}↓ → ιx φ{x} = ιx(𝒜φ{x})›
1826proof(rule "→I")
1827  AOT_assume ‹ιx φ{x}↓›
1828  AOT_hence ‹∃y (y = ιx φ{x})› by (metis "rule=I:1" "existential:1")
1829  then AOT_obtain a where ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1830  moreover AOT_have ‹a = ιx(𝒜φ{x})› using calculation "equi-desc-descA:1"[THEN "≡E"(1)] by blast
1831  ultimately AOT_show ‹ιx φ{x} = ιx(𝒜φ{x})› using "rule=E" by fast
1832qed
1833
1834AOT_theorem "nec-hintikka-scheme": ‹x = ιx φ{x} ≡ 𝒜φ{x} & ∀z(𝒜φ{z} → z = x)›
1835proof -
1836  AOT_have ‹x = ιx φ{x} ≡ ∀z(𝒜φ{z} ≡ z = x)› using descriptions[axiom_inst] by blast
1837  also AOT_have ‹… ≡ (𝒜φ{x} & ∀z(𝒜φ{z} → z = x))›
1838    using "Commutativity of ≡"[THEN "≡E"(1)] "term-out:3" by fast
1839  finally show ?thesis.
1840qed
1841
1842AOT_theorem "equiv-desc-eq:1": ‹𝒜∀x(φ{x} ≡ ψ{x}) → ∀x (x = ιx φ{x} ≡ x = ιx ψ{x})›
1843proof(rule "→I"; rule "∀I")
1844  fix β
1845  AOT_assume ‹𝒜∀x(φ{x} ≡ ψ{x})›
1846  AOT_hence ‹𝒜(φ{x} ≡ ψ{x})› for x using "logic-actual-nec:3"[axiom_inst, THEN "≡E"(1)] "∀E"(2) by blast
1847  AOT_hence 0: ‹𝒜φ{x} ≡ 𝒜ψ{x}› for x by (metis "Act-Basic:5" "≡E"(1))
1848  AOT_have ‹β = ιx φ{x} ≡ 𝒜φ{β} & ∀z(𝒜φ{z} → z = β)› using "nec-hintikka-scheme" by blast
1849  also AOT_have ‹... ≡ 𝒜ψ{β} & ∀z(𝒜ψ{z} → z = β)›
1850  proof (rule "≡I"; rule "→I")
1851    AOT_assume 1: ‹𝒜φ{β} & ∀z(𝒜φ{z} → z = β)›
1852    AOT_hence ‹𝒜φ{z} → z = β› for z using "&E" "∀E" by blast
1853    AOT_hence ‹𝒜ψ{z} → z = β› for z using 0 "≡E" "→I" "→E" by metis
1854    AOT_hence ‹∀z(𝒜ψ{z} → z = β)› using "∀I" by fast
1855    moreover AOT_have ‹𝒜ψ{β}› using "&E" 0[THEN "≡E"(1)] 1 by blast
1856    ultimately AOT_show ‹𝒜ψ{β} & ∀z(𝒜ψ{z} → z = β)› using "&I" by blast
1857  next
1858    AOT_assume 1: ‹𝒜ψ{β} & ∀z(𝒜ψ{z} → z = β)›
1859    AOT_hence ‹𝒜ψ{z} → z = β› for z using "&E" "∀E" by blast
1860    AOT_hence ‹𝒜φ{z} → z = β› for z using 0 "≡E" "→I" "→E" by metis
1861    AOT_hence ‹∀z(𝒜φ{z} → z = β)› using "∀I" by fast
1862    moreover AOT_have ‹𝒜φ{β}› using "&E" 0[THEN "≡E"(2)] 1 by blast
1863    ultimately AOT_show ‹𝒜φ{β} & ∀z(𝒜φ{z} → z = β)› using "&I" by blast
1864  qed
1865  also AOT_have ‹... ≡ β = ιx ψ{x}›
1866    using "Commutativity of ≡"[THEN "≡E"(1)] "nec-hintikka-scheme" by blast
1867  finally AOT_show ‹β = ιx φ{x} ≡ β = ιx ψ{x}› .
1868qed
1869
1870AOT_theorem "equiv-desc-eq:2": ‹ιx φ{x}↓ & 𝒜∀x(φ{x} ≡ ψ{x}) → ιx φ{x} = ιx ψ{x}›
1871proof(rule "→I")
1872  AOT_assume ‹ιx φ{x}↓ & 𝒜∀x(φ{x} ≡ ψ{x})›
1873  AOT_hence 0: ‹∃y (y = ιx φ{x})› and
1874            1: ‹∀x (x = ιx φ{x} ≡ x = ιx ψ{x})›
1875    using "&E" "free-thms:1"[THEN "≡E"(1)] "equiv-desc-eq:1" "→E" by blast+
1876  then AOT_obtain a where ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1877  moreover AOT_have ‹a = ιx ψ{x}› using calculation 1 "∀E" "≡E"(1) by fast
1878  ultimately AOT_show ‹ιx φ{x} = ιx ψ{x}›
1879    using "rule=E" by fast
1880qed
1881
1882AOT_theorem "equiv-desc-eq:3": ‹ιx φ{x}↓ & □∀x(φ{x} ≡ ψ{x}) → ιx φ{x} = ιx ψ{x}›
1883  using "→I" "equiv-desc-eq:2"[THEN "→E", OF "&I"] "&E" "nec-imp-act"[THEN "→E"] by metis
1884
1885(* Note: this is a special case of "exist-nec" *)
1886AOT_theorem "equiv-desc-eq:4": ‹ιx φ{x}↓ → □ιx φ{x}↓›
1887proof(rule "→I")
1888  AOT_assume ‹ιx φ{x}↓›
1889  AOT_hence ‹∃y (y = ιx φ{x})› by (metis "rule=I:1" "existential:1")
1890  then AOT_obtain a where ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1891  AOT_thus ‹□ιx φ{x}↓›
1892    using "ex:2:a" "rule=E" by fast
1893qed
1894
1895AOT_theorem "equiv-desc-eq:5": ‹ιx φ{x}↓ → ∃y □(y = ιx φ{x})›
1896proof(rule "→I")
1897  AOT_assume ‹ιx φ{x}↓›
1898  AOT_hence ‹∃y (y = ιx φ{x})› by (metis "rule=I:1" "existential:1")
1899  then AOT_obtain a where ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1900  AOT_hence ‹□(a = ιx φ{x})› by (metis "id-nec:2" "vdash-properties:10")
1901  AOT_thus ‹∃y □(y = ιx φ{x})› by (rule "∃I")
1902qed
1903
1904AOT_act_theorem "equiv-desc-eq2:1": ‹∀x (φ{x} ≡ ψ{x}) → ∀x (x = ιx φ{x} ≡ x = ιx ψ{x})›
1905  using "→I" "logic-actual"[act_axiom_inst, THEN "→E"] "equiv-desc-eq:1"[THEN "→E"]
1906        "RA[1]" "deduction-theorem" by blast
1907
1908AOT_act_theorem "equiv-desc-eq2:2": ‹ιx φ{x}↓ & ∀x (φ{x} ≡ ψ{x}) → ιx φ{x} = ιx ψ{x}›
1909  using "→I" "logic-actual"[act_axiom_inst, THEN "→E"] "equiv-desc-eq:2"[THEN "→E", OF "&I"]
1910        "RA[1]" "deduction-theorem" "&E" by metis
1911
1912context russel_axiom
1913begin
1914AOT_theorem "nec-russell-axiom": ‹ψ{ιx φ{x}} ≡ ∃x(𝒜φ{x} & ∀z(𝒜φ{z} → z = x) & ψ{x})›
1915proof -
1916  AOT_have b: ‹∀x (x = ιx φ{x} ≡ (𝒜φ{x} & ∀z(𝒜φ{z} → z = x)))›
1917    using "nec-hintikka-scheme" "∀I" by fast
1918  show ?thesis
1919  proof(rule "≡I"; rule "→I")
1920    AOT_assume c: ‹ψ{ιx φ{x}}›
1921    AOT_hence d: ‹ιx φ{x}↓› using ψ_denotes_asm by blast
1922    AOT_hence ‹∃y (y = ιx φ{x})› by (metis "rule=I:1" "existential:1")
1923    then AOT_obtain a where a_def: ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1924    moreover AOT_have ‹a = ιx φ{x} ≡ (𝒜φ{a} & ∀z(𝒜φ{z} → z = a))› using b "∀E" by blast
1925    ultimately AOT_have ‹𝒜φ{a} & ∀z(𝒜φ{z} → z = a)› using "≡E" by blast
1926    moreover AOT_have ‹ψ{a}›
1927    proof - 
1928      AOT_have 1: ‹∀x∀y(x = y → y = x)›
1929        by (simp add: "id-eq:2" "universal-cor")
1930      AOT_have ‹a = ιx φ{x} →  ιx φ{x} = a›
1931        by (rule "∀E"(1)[where τ="«ιx φ{x}»"]; rule "∀E"(2)[where β=a])
1932           (auto simp: d "universal-cor" 1)
1933      AOT_thus ‹ψ{a}›
1934        using a_def c "rule=E" "→E" by metis
1935    qed
1936    ultimately AOT_have ‹𝒜φ{a} & ∀z(𝒜φ{z} → z = a) & ψ{a}› by (rule "&I")
1937    AOT_thus ‹∃x(𝒜φ{x} & ∀z(𝒜φ{z} → z = x) & ψ{x})› by (rule "∃I")
1938  next
1939    AOT_assume ‹∃x(𝒜φ{x} & ∀z(𝒜φ{z} → z = x) & ψ{x})›
1940    then AOT_obtain b where g: ‹𝒜φ{b} & ∀z(𝒜φ{z} → z = b) & ψ{b}› using "instantiation"[rotated] by blast
1941    AOT_hence h: ‹b = ιx φ{x} ≡ (𝒜φ{b} & ∀z(𝒜φ{z} → z = b))› using b "∀E" by blast
1942    AOT_have ‹𝒜φ{b} & ∀z(𝒜φ{z} → z = b)› and j: ‹ψ{b}› using g "&E" by blast+
1943    AOT_hence ‹b = ιx φ{x}› using h "≡E" by blast
1944    AOT_thus ‹ψ{ιx φ{x}}› using j "rule=E" by blast
1945  qed
1946qed
1947end
1948
1949AOT_theorem "actual-desc:1": ‹ιx φ{x}↓ ≡ ∃!x 𝒜φ{x}›
1950proof (rule "≡I"; rule "→I")
1951  AOT_assume ‹ιx φ{x}↓›
1952  AOT_hence ‹∃y (y = ιx φ{x})› by (metis "rule=I:1" "existential:1")
1953  then AOT_obtain a where ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1954  moreover AOT_have ‹a = ιx φ{x} ≡ ∀z(𝒜φ{z} ≡ z = a)›
1955    using descriptions[axiom_inst] by blast
1956  ultimately AOT_have ‹∀z(𝒜φ{z} ≡ z = a)›
1957    using "≡E" by blast
1958  AOT_hence ‹∃x∀z(𝒜φ{z} ≡ z = x)› by (rule "∃I")
1959  AOT_thus ‹∃!x 𝒜φ{x}›
1960    using "uniqueness:2"[THEN "≡E"(2)] by fast
1961next
1962  AOT_assume ‹∃!x 𝒜φ{x}›
1963  AOT_hence ‹∃x∀z(𝒜φ{z} ≡ z = x)›
1964    using "uniqueness:2"[THEN "≡E"(1)] by fast
1965  then AOT_obtain a where ‹∀z(𝒜φ{z} ≡ z = a)› using "instantiation"[rotated] by blast
1966  moreover AOT_have ‹a = ιx φ{x} ≡ ∀z(𝒜φ{z} ≡ z = a)›
1967    using descriptions[axiom_inst] by blast
1968  ultimately AOT_have ‹a = ιx φ{x}› using "≡E" by blast
1969  AOT_thus ‹ιx φ{x}↓› by (metis "t=t-proper:2" "vdash-properties:6")
1970qed
1971
1972AOT_theorem "actual-desc:2": ‹x = ιx φ{x} → 𝒜φ{x}›
1973  using "&E"(1) "contraposition:1[2]" "≡E"(1) "nec-hintikka-scheme" "reductio-aa:2" "vdash-properties:9" by blast
1974
1975AOT_theorem "actual-desc:3": ‹z = ιx φ{x} → 𝒜φ{z}›
1976  using "actual-desc:2". (* TODO: same as above *)
1977
1978AOT_theorem "actual-desc:4": ‹ιx φ{x}↓ → 𝒜φ{ιx φ{x}}›
1979proof(rule "→I")
1980  AOT_assume ‹ιx φ{x}↓›
1981  AOT_hence ‹∃y (y = ιx φ{x})› by (metis "rule=I:1" "existential:1")
1982  then AOT_obtain a where ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1983  AOT_thus ‹𝒜φ{ιx φ{x}}›
1984    using "actual-desc:2" "rule=E" "→E" by fast
1985qed
1986
1987(* TODO: take another look at proof in PLM *)
1988AOT_theorem "actual-desc:5": ‹ιx φ{x} = ιx ψ{x} → 𝒜∀x(φ{x} ≡ ψ{x})›
1989proof(rule "→I")
1990  AOT_assume 0: ‹ιx φ{x} = ιx ψ{x}›
1991  AOT_hence φ_down: ‹ιx φ{x}↓› and ψ_down: ‹ιx ψ{x}↓›
1992    using "t=t-proper:1" "t=t-proper:2" "vdash-properties:6" by blast+
1993  AOT_hence ‹∃y (y = ιx φ{x})› and ‹∃y (y = ιx ψ{x})› by (metis "rule=I:1" "existential:1")+
1994  then AOT_obtain a and b where a_eq: ‹a = ιx φ{x}› and b_eq: ‹b = ιx ψ{x}›
1995    using "instantiation"[rotated] by metis
1996
1997  AOT_have ‹∀α∀β (α = β → β = α)› by (rule "∀I"; rule "∀I"; rule "id-eq:2")
1998  AOT_hence ‹∀β (ιx φ{x} = β → β = ιx φ{x})›
1999    using "∀E" φ_down by blast
2000  AOT_hence ‹ιx φ{x} = ιx ψ{x} → ιx ψ{x} = ιx φ{x}›
2001    using "∀E" ψ_down by blast
2002  AOT_hence 1: ‹ιx ψ{x} = ιx φ{x}› using 0
2003    "→E" by blast
2004
2005  AOT_have ‹𝒜φ{x} ≡ 𝒜ψ{x}› for x
2006  proof(rule "≡I"; rule "→I")
2007    AOT_assume ‹𝒜φ{x}›
2008    moreover AOT_have ‹𝒜φ{x} → x = a› for x
2009      using "nec-hintikka-scheme"[THEN "≡E"(1), OF a_eq, THEN "&E"(2)] "∀E" by blast
2010    ultimately AOT_have ‹x = a› using "→E" by blast
2011    AOT_hence ‹x = ιx φ{x}› using a_eq "rule=E" by blast
2012    AOT_hence ‹x = ιx ψ{x}› using 0 "rule=E" by blast
2013    AOT_thus ‹𝒜ψ{x}› by (metis "actual-desc:3" "vdash-properties:6")
2014  next
2015    AOT_assume ‹𝒜ψ{x}›
2016    moreover AOT_have ‹𝒜ψ{x} → x = b› for x
2017      using "nec-hintikka-scheme"[THEN "≡E"(1), OF b_eq, THEN "&E"(2)] "∀E" by blast
2018    ultimately AOT_have ‹x = b› using "→E" by blast
2019    AOT_hence ‹x = ιx ψ{x}› using b_eq "rule=E" by blast
2020    AOT_hence ‹x = ιx φ{x}› using 1 "rule=E" by blast
2021    AOT_thus ‹𝒜φ{x}› by (metis "actual-desc:3" "vdash-properties:6")
2022  qed
2023  AOT_hence ‹𝒜(φ{x} ≡ ψ{x})› for x by (metis "Act-Basic:5" "≡E"(2))
2024  AOT_hence ‹∀x 𝒜(φ{x} ≡ ψ{x})› by (rule "∀I")
2025  AOT_thus ‹𝒜∀x (φ{x} ≡ ψ{x})›
2026    using "logic-actual-nec:3"[axiom_inst, THEN "≡E"(2)] by fast
2027qed    
2028
2029AOT_theorem "!box-desc:1": ‹∃!x □φ{x} → ∀y (y = ιx φ{x} → φ{y})›
2030proof(rule "→I")
2031  AOT_assume ‹∃!x □φ{x}›
2032  AOT_hence ζ: ‹∃x (□φ{x} & ∀z (□φ{z} → z = x))›
2033    using "uniqueness:1"[THEN "≡dfE"] by blast
2034  then AOT_obtain b where θ: ‹□φ{b} & ∀z (□φ{z} → z = b)› using "instantiation"[rotated] by blast
2035  AOT_show ‹∀y (y = ιx φ{x} → φ{y})›
2036  proof(rule GEN; rule "→I")
2037    fix y
2038    AOT_assume ‹y = ιx φ{x}›
2039    AOT_hence ‹𝒜φ{y} & ∀z (𝒜φ{z} → z = y)› using "nec-hintikka-scheme"[THEN "≡E"(1)] by blast
2040    AOT_hence ‹𝒜φ{b} → b = y› using "&E" "∀E" by blast
2041    moreover AOT_have ‹𝒜φ{b}› using θ[THEN "&E"(1)]  by (metis "nec-imp-act" "→E")
2042    ultimately AOT_have ‹b = y› using "→E" by blast
2043    moreover AOT_have ‹φ{b}› using θ[THEN "&E"(1)]  by (metis "qml:2"[axiom_inst] "→E") 
2044    ultimately AOT_show ‹φ{y}› using "rule=E" by blast
2045  qed
2046qed
2047
2048AOT_theorem "!box-desc:2": ‹∀x (φ{x} → □φ{x}) → (∃!x φ{x} → ∀y (y = ιx φ{x} → φ{y}))›
2049proof(rule "→I"; rule "→I")
2050  AOT_assume ‹∀x (φ{x} → □φ{x})›
2051  moreover AOT_assume ‹∃!x φ{x}›
2052  ultimately AOT_have ‹∃!x □φ{x}›
2053    using "nec-exist-!"[THEN "→E", THEN "→E"] by blast
2054  AOT_thus ‹∀y (y = ιx φ{x} → φ{y})›
2055    using "!box-desc:1" "→E" by blast
2056qed
2057
2058AOT_theorem "dr-alphabetic-thm": ‹ιν φ{ν}↓ → ιν φ{ν} = ιμ φ{μ}› (* TODO: vacuous *)
2059  by (simp add: "rule=I:1" "→I")
2060
2061AOT_theorem "RM:1[prem]": assumes ‹Γ ⊢□ φ → ψ› shows ‹□Γ ⊢□ □φ → □ψ›
2062proof -
2063  AOT_have ‹□Γ ⊢□ □(φ → ψ)› using "RN[prem]" assms by blast
2064  AOT_thus ‹□Γ ⊢□ □φ → □ψ› by (metis "qml:1"[axiom_inst] "→E")
2065qed
2066
2067AOT_theorem "RM:1": assumes ‹⊢□ φ → ψ› shows ‹⊢□ □φ → □ψ›
2068  using "RM:1[prem]" assms by blast
2069
2070lemmas RM = "RM:1"
2071
2072AOT_theorem "RM:2[prem]": assumes ‹Γ ⊢□ φ → ψ› shows ‹□Γ ⊢□ ◇φ → ◇ψ›
2073proof -
2074  AOT_have ‹Γ ⊢□ ¬ψ → ¬φ› using assms 
2075    by (simp add: "contraposition:1[1]")
2076  AOT_hence ‹□Γ ⊢□ □¬ψ → □¬φ› using "RM:1[prem]" by blast
2077  AOT_thus ‹□Γ ⊢□ ◇φ → ◇ψ›
2078    by (meson "≡dfE" "≡dfI" "conventions:5" "deduction-theorem" "modus-tollens:1")
2079qed
2080
2081AOT_theorem "RM:2": assumes ‹⊢□ φ → ψ› shows ‹⊢□ ◇φ → ◇ψ›
2082  using "RM:2[prem]" assms by blast
2083
2084lemmas "RM◇" = "RM:2"
2085
2086AOT_theorem "RM:3[prem]": assumes ‹Γ ⊢□ φ ≡ ψ› shows ‹□Γ ⊢□ □φ ≡ □ψ›
2087proof -
2088  AOT_have ‹Γ ⊢□ φ → ψ› and ‹Γ ⊢□ ψ → φ› using assms "≡E" "→I" by metis+
2089  AOT_hence ‹□Γ ⊢□ □φ → □ψ› and ‹□Γ ⊢□ □ψ → □φ› using "RM:1[prem]" by metis+
2090  AOT_thus ‹□Γ ⊢□ □φ ≡ □ψ›
2091    by (simp add: "≡I")
2092qed
2093
2094AOT_theorem "RM:3": assumes ‹⊢□ φ ≡ ψ› shows ‹⊢□ □φ ≡ □ψ›
2095  using "RM:3[prem]" assms by blast
2096
2097lemmas RE = "RM:3"
2098
2099AOT_theorem "RM:4[prem]": assumes ‹Γ ⊢□ φ ≡ ψ› shows ‹□Γ ⊢□ ◇φ ≡ ◇ψ›
2100proof -
2101  AOT_have ‹Γ ⊢□ φ → ψ› and ‹Γ ⊢□ ψ → φ› using assms "≡E" "→I" by metis+
2102  AOT_hence ‹□Γ ⊢□ ◇φ → ◇ψ› and ‹□Γ ⊢□ ◇ψ → ◇φ› using "RM:2[prem]" by metis+
2103  AOT_thus ‹□Γ ⊢□ ◇φ ≡ ◇ψ› by (simp add: "≡I")
2104qed
2105
2106AOT_theorem "RM:4": assumes ‹⊢□ φ ≡ ψ› shows ‹⊢□ ◇φ ≡ ◇ψ›
2107  using "RM:4[prem]" assms by blast
2108
2109lemmas "RE◇" = "RM:4"
2110
2111AOT_theorem "KBasic:1": ‹□φ → □(ψ → φ)›
2112  by (simp add: RM "pl:1"[axiom_inst])
2113
2114AOT_theorem "KBasic:2": ‹□¬φ → □(φ → ψ)›
2115  by (simp add: RM "useful-tautologies:3")
2116
2117AOT_theorem "KBasic:3": ‹□(φ & ψ) ≡ (□φ & □ψ)›
2118proof (rule "≡I"; rule "→I")
2119  AOT_assume ‹□(φ & ψ)›
2120  AOT_thus ‹□φ & □ψ›
2121    by (meson RM "&I" "Conjunction Simplification"(1) "Conjunction Simplification"(2) "vdash-properties:6")
2122next
2123  AOT_have ‹□φ → □(ψ → (φ & ψ))› by (simp add: "RM:1" Adjunction)
2124  AOT_hence ‹□φ → (□ψ → □(φ & ψ))›  by (metis "Hypothetical Syllogism" "qml:1"[axiom_inst])
2125  moreover AOT_assume ‹□φ & □ψ›
2126  ultimately AOT_show ‹□(φ & ψ)›
2127    using "→E" "&E" by blast
2128qed
2129
2130AOT_theorem "KBasic:4": ‹□(φ ≡ ψ) ≡ (□(φ → ψ) & □(ψ → φ))›
2131proof -
2132  AOT_have θ: ‹□((φ → ψ) & (ψ → φ)) ≡ (□(φ → ψ) & □(ψ → φ))›
2133    by (fact "KBasic:3")
2134  AOT_modally_strict {
2135    AOT_have ‹(φ ≡ ψ) ≡ ((φ → ψ) & (ψ → φ))›
2136      by (fact "conventions:3"[THEN "≡Df"])
2137  }
2138  AOT_hence ξ: ‹□(φ ≡ ψ) ≡ □((φ → ψ) & (ψ → φ))›
2139    by (rule RE)
2140  with ξ and θ AOT_show ‹□(φ ≡ ψ) ≡ (□(φ → ψ) & □(ψ → φ))›
2141    using "≡E"(5) by blast
2142qed
2143
2144AOT_theorem "KBasic:5": ‹(□(φ → ψ) & □(ψ → φ)) → (□φ ≡ □ψ)›
2145proof -
2146  AOT_have ‹□(φ → ψ) → (□φ → □ψ)›
2147    by (fact "qml:1"[axiom_inst])
2148  moreover AOT_have ‹□(ψ → φ) → (□ψ → □φ)›
2149    by (fact "qml:1"[axiom_inst])
2150  ultimately AOT_have ‹(□(φ → ψ) & □(ψ → φ)) → ((□φ → □ψ) & (□ψ → □φ))›
2151    by (metis "&I" MP "Double Composition")
2152  moreover AOT_have ‹((□φ → □ψ) & (□ψ → □φ)) → (□φ ≡ □ψ)›
2153    using "conventions:3"[THEN "≡dfI"] "→I" by blast
2154  ultimately AOT_show ‹(□(φ → ψ) & □(ψ → φ)) → (□φ ≡ □ψ)›
2155    by (metis "Hypothetical Syllogism")
2156qed
2157
2158AOT_theorem "KBasic:6": ‹□(φ≡ ψ) → (□φ ≡ □ψ)›
2159  using "KBasic:4" "KBasic:5" "deduction-theorem" "≡E"(1) "vdash-properties:10" by blast
2160AOT_theorem "KBasic:7": ‹((□φ & □ψ) ∨ (□¬φ & □¬ψ)) → □(φ ≡ ψ)›
2161proof (rule "→I"; drule "∨E"(1); (rule "→I")?)
2162  AOT_assume ‹□φ & □ψ›
2163  AOT_hence ‹□φ› and ‹□ψ› using "&E" by blast+
2164  AOT_hence ‹□(φ → ψ)› and ‹□(ψ → φ)› using "KBasic:1" "→E" by blast+
2165  AOT_hence ‹□(φ → ψ) & □(ψ → φ)› using "&I" by blast
2166  AOT_thus ‹□(φ ≡ ψ)›  by (metis "KBasic:4" "≡E"(2))
2167next
2168  AOT_assume ‹□¬φ & □¬ψ›
2169  AOT_hence 0: ‹□(¬φ & ¬ψ)› using "KBasic:3"[THEN "≡E"(2)] by blast
2170  AOT_modally_strict {
2171    AOT_have ‹(¬φ & ¬ψ) → (φ ≡ ψ)›
2172      by (metis "&E"(1) "&E"(2) "deduction-theorem" "≡I" "reductio-aa:1")
2173  }
2174  AOT_hence ‹□(¬φ & ¬ψ) → □(φ ≡ ψ)›
2175    by (rule RM)
2176  AOT_thus ‹□(φ ≡ ψ)› using 0 "→E" by blast
2177qed(auto)
2178
2179AOT_theorem "KBasic:8": ‹□(φ & ψ) → □(φ ≡ ψ)›
2180  by (meson "RM:1" "&E"(1) "&E"(2) "deduction-theorem" "≡I")
2181AOT_theorem "KBasic:9": ‹□(¬φ & ¬ψ) → □(φ ≡ ψ)›
2182  by (metis "RM:1" "&E"(1) "&E"(2) "deduction-theorem" "≡I" "raa-cor:4")
2183AOT_theorem "KBasic:10": ‹□φ ≡ □¬¬φ›
2184  by (simp add: "RM:3" "oth-class-taut:3:b")
2185AOT_theorem "KBasic:11": ‹¬□φ ≡ ◇¬φ›
2186proof (rule "≡I"; rule "→I")
2187  AOT_show ‹◇¬φ› if ‹¬□φ›
2188    using that "≡dfI" "conventions:5" "KBasic:10" "≡E"(3) by blast
2189next
2190  AOT_show ‹¬□φ› if ‹◇¬φ›
2191    using "≡dfE" "conventions:5" "KBasic:10" "≡E"(4) that by blast
2192qed
2193AOT_theorem "KBasic:12": ‹□φ ≡ ¬◇¬φ›
2194proof (rule "≡I"; rule "→I")
2195  AOT_show ‹¬◇¬φ› if ‹□φ›
2196    using "¬¬I" "KBasic:11" "≡E"(3) that by blast
2197next
2198  AOT_show ‹□φ› if ‹¬◇¬φ›
2199  using "KBasic:11" "≡E"(1) "reductio-aa:1" that by blast
2200qed
2201AOT_theorem "KBasic:13": ‹□(φ → ψ) → (◇φ → ◇ψ)›
2202proof -
2203  AOT_have ‹φ → ψ ⊢□ φ → ψ› by blast
2204  AOT_hence ‹□(φ → ψ) ⊢□ ◇φ → ◇ψ›
2205    using "RM:2[prem]" by blast
2206  AOT_thus ‹□(φ → ψ) → (◇φ → ◇ψ)› using "→I" by blast
2207qed
2208lemmas "K◇" = "KBasic:13"
2209AOT_theorem "KBasic:14": ‹◇□φ ≡ ¬□◇¬φ›
2210  by (meson "RE◇" "KBasic:11" "KBasic:12" "≡E"(6) "oth-class-taut:3:a")
2211AOT_theorem "KBasic:15": ‹(□φ ∨ □ψ) → □(φ ∨ ψ)›
2212proof -
2213  AOT_modally_strict {
2214    AOT_have ‹φ → (φ ∨ ψ)› and ‹ψ → (φ ∨ ψ)›
2215      by (auto simp: "Disjunction Addition"(1) "Disjunction Addition"(2))
2216  }
2217  AOT_hence ‹□φ → □(φ ∨ ψ)› and ‹□ψ → □(φ ∨ ψ)›
2218    using RM by blast+
2219  AOT_thus ‹(□φ ∨ □ψ) → □(φ ∨ ψ)›
2220    by (metis "∨E"(1) "deduction-theorem")
2221qed
2222
2223AOT_theorem "KBasic:16": ‹(□φ & ◇ψ) → ◇(φ & ψ)›
2224  by (meson "KBasic:13" "RM:1" Adjunction "Hypothetical Syllogism" Importation "vdash-properties:6")
2225
2226AOT_theorem "rule-sub-lem:1:a":
2227  assumes ‹⊢□ □(ψ ≡ χ)›
2228  shows ‹⊢□ ¬ψ ≡ ¬χ›
2229  using "qml:2"[axiom_inst, THEN "→E", OF assms]
2230        "≡E"(1) "oth-class-taut:4:b" by blast
2231
2232AOT_theorem "rule-sub-lem:1:b":
2233  assumes ‹⊢□ □(ψ ≡ χ)›
2234  shows ‹⊢□ (ψ → Θ) ≡ (χ → Θ)›
2235  using "qml:2"[axiom_inst, THEN "→E", OF assms]
2236  using "oth-class-taut:4:c" "vdash-properties:6" by blast
2237
2238AOT_theorem "rule-sub-lem:1:c":
2239  assumes ‹⊢□ □(ψ ≡ χ)›
2240  shows ‹⊢□ (Θ → ψ) ≡ (Θ → χ)›
2241  using "qml:2"[axiom_inst, THEN "→E", OF assms]
2242  using "oth-class-taut:4:d" "vdash-properties:6" by blast
2243
2244AOT_theorem "rule-sub-lem:1:d":
2245  assumes ‹for arbitrary α: ⊢□ □(ψ{α} ≡ χ{α})›
2246  shows ‹⊢□ ∀α ψ{α} ≡ ∀α χ{α}›
2247proof -
2248  AOT_modally_strict {
2249    AOT_have ‹∀α (ψ{α} ≡ χ{α})›
2250      using "qml:2"[axiom_inst, THEN "→E", OF assms] "∀I" by fast
2251    AOT_hence 0: ‹ψ{α} ≡ χ{α}› for α using "∀E" by blast
2252    AOT_show ‹∀α ψ{α} ≡ ∀α χ{α}›
2253    proof (rule "≡I"; rule "→I")
2254      AOT_assume ‹∀α ψ{α}›
2255      AOT_hence ‹ψ{α}› for α using "∀E" by blast
2256      AOT_hence ‹χ{α}› for α using 0 "≡E" by blast
2257      AOT_thus ‹∀α χ{α}› by (rule "∀I")
2258    next
2259      AOT_assume ‹∀α χ{α}›
2260      AOT_hence ‹χ{α}› for α using "∀E" by blast
2261      AOT_hence ‹ψ{α}› for α using 0 "≡E" by blast
2262      AOT_thus ‹∀α ψ{α}› by (rule "∀I")
2263    qed
2264  }
2265qed
2266
2267AOT_theorem "rule-sub-lem:1:e":
2268  assumes ‹⊢□ □(ψ ≡ χ)›
2269  shows ‹⊢□ [λ ψ] ≡ [λ χ]›
2270  using "qml:2"[axiom_inst, THEN "→E", OF assms]
2271  using "≡E"(1) "propositions-lemma:6" by blast
2272
2273AOT_theorem "rule-sub-lem:1:f":
2274  assumes ‹⊢□ □(ψ ≡ χ)›
2275  shows ‹⊢□ 𝒜ψ ≡ 𝒜χ›
2276  using "qml:2"[axiom_inst, THEN "→E", OF assms, THEN "RA[2]"]
2277  by (metis "Act-Basic:5" "≡E"(1))
2278
2279AOT_theorem "rule-sub-lem:1:g":
2280  assumes ‹⊢□ □(ψ ≡ χ)›
2281  shows ‹⊢□ □ψ ≡ □χ›
2282  using "KBasic:6" assms "vdash-properties:6" by blast
2283
2284text‹Note that instead of deriving @{text "rule-sub-lem:2"}, @{text "rule-sub-lem:3"}, @{text "rule-sub-lem:4"},
2285     and @{text "rule-sub-nec"}, we construct substitution methods instead.›
2286
2287class AOT_subst =
2288  fixes AOT_subst :: "('a ⇒ 𝗈) ⇒ bool"
2289    and AOT_subst_cond :: "'a ⇒ 'a ⇒ bool"
2290  assumes AOT_subst: "AOT_subst φ ⟹ AOT_subst_cond ψ χ ⟹ [v ⊨ «φ ψ» ≡ «φ χ»]"
2291
2292named_theorems AOT_substI
2293
2294instantiation 𝗈 :: AOT_subst
2295begin
2296
2297inductive AOT_subst_𝗈 where
2298  AOT_subst_𝗈_id[AOT_substI]: "AOT_subst_𝗈 (λφ. φ)"
2299| AOT_subst_𝗈_const[AOT_substI]: "AOT_subst_𝗈 (λφ. ψ)"
2300| AOT_subst_𝗈_not[AOT_substI]: "AOT_subst_𝗈 Θ ⟹ AOT_subst_𝗈 (λ φ. «¬Θ{φ}»)"
2301| AOT_subst_𝗈_imp[AOT_substI]: "AOT_subst_𝗈 Θ ⟹ AOT_subst_𝗈 Ξ ⟹ AOT_subst_𝗈 (λ φ. «Θ{φ} → Ξ{φ}»)"
2302| AOT_subst_𝗈_lambda0[AOT_substI]: "AOT_subst_𝗈 Θ ⟹ AOT_subst_𝗈 (λ φ. (AOT_lambda0 (Θ φ)))"
2303| AOT_subst_𝗈_act[AOT_substI]: "AOT_subst_𝗈 Θ ⟹ AOT_subst_𝗈 (λ φ. «𝒜Θ{φ}»)"
2304| AOT_subst_𝗈_box[AOT_substI]: "AOT_subst_𝗈 Θ ⟹ AOT_subst_𝗈 (λ φ. «□Θ{φ}»)"
2305| AOT_subst_𝗈_by_def[AOT_substI]: "(⋀ ψ . AOT_model_equiv_def (Θ ψ) (Ξ ψ)) ⟹ AOT_subst_𝗈 Ξ ⟹ AOT_subst_𝗈 Θ"
2306
2307definition AOT_subst_cond_𝗈 where "AOT_subst_cond_𝗈 ≡ λ ψ χ . ∀ v . [v ⊨ ψ ≡ χ]"
2308
2309instance
2310proof
2311  fix ψ χ :: 𝗈 and φ :: ‹𝗈 ⇒ 𝗈›
2312  assume cond: ‹AOT_subst_cond ψ χ›
2313  assume ‹AOT_subst φ›
2314  moreover AOT_have ‹⊢□ ψ ≡ χ› using cond unfolding AOT_subst_cond_𝗈_def by blast
2315  ultimately AOT_show ‹⊢□ φ{ψ} ≡ φ{χ}›
2316  proof (induct arbitrary: ψ χ)
2317    case AOT_subst_𝗈_id
2318    thus ?case using "≡E"(2) "oth-class-taut:4:b" "rule-sub-lem:1:a" by blast
2319  next
2320    case (AOT_subst_𝗈_const ψ)
2321    thus ?case by (simp add: "oth-class-taut:3:a")
2322  next
2323    case (AOT_subst_𝗈_not Θ)
2324    thus ?case by (simp add: RN "rule-sub-lem:1:a")
2325  next
2326    case (AOT_subst_𝗈_imp Θ Ξ)
2327    thus ?case by (meson RN "≡E"(5) "rule-sub-lem:1:b" "rule-sub-lem:1:c")
2328  next
2329    case (AOT_subst_𝗈_lambda0 Θ)
2330    thus ?case by (simp add: RN "rule-sub-lem:1:e")
2331  next
2332    case (AOT_subst_𝗈_act Θ)
2333    thus ?case by (simp add: RN "rule-sub-lem:1:f")
2334  next
2335    case (AOT_subst_𝗈_box Θ)
2336    thus ?case by (simp add: RN "rule-sub-lem:1:g")
2337  next
2338    case (AOT_subst_𝗈_by_def Θ Ξ)
2339    AOT_modally_strict {
2340      AOT_have ‹Ξ{ψ} ≡ Ξ{χ}› using AOT_subst_𝗈_by_def by simp
2341      AOT_thus ‹Θ{ψ} ≡ Θ{χ}›
2342        using "≡Df"[OF AOT_subst_𝗈_by_def(1), of _ ψ] "≡Df"[OF AOT_subst_𝗈_by_def(1), of _ χ]
2343        by (metis "≡E"(6) "oth-class-taut:3:a")
2344    }
2345  qed
2346qed
2347end
2348
2349instantiation "fun" :: (AOT_Term_id_2, AOT_subst) AOT_subst
2350begin
2351
2352definition AOT_subst_cond_fun :: "('a ⇒ 'b) ⇒ ('a ⇒ 'b) ⇒ bool" where
2353  "AOT_subst_cond_fun ≡ λ φ ψ . ∀ α . AOT_subst_cond (φ (AOT_term_of_var α)) (ψ (AOT_term_of_var α))"
2354
2355inductive AOT_subst_fun :: "(('a ⇒ 'b) ⇒ 𝗈) ⇒ bool" where
2356  AOT_subst_fun_const[AOT_substI]: "AOT_subst_fun (λφ. ψ)"
2357| AOT_subst_fun_id[AOT_substI]: "AOT_subst Ψ ⟹ AOT_subst_fun (λφ. Ψ (φ (AOT_term_of_var x)))"
2358| AOT_subst_fun_all[AOT_substI]: "AOT_subst Ψ ⟹ (⋀ α . AOT_subst_fun (Θ (AOT_term_of_var α))) ⟹ AOT_subst_fun (λφ :: 'a ⇒ 'b. Ψ «∀α «Θ (α::'a) φ»»)"
2359| AOT_subst_fun_not[AOT_substI]: "AOT_subst Ψ ⟹ AOT_subst_fun (λφ. «¬«Ψ φ»»)"
2360| AOT_subst_fun_imp[AOT_substI]: "AOT_subst Ψ ⟹ AOT_subst Θ ⟹ AOT_subst_fun (λφ. ««Ψ φ» → «Θ φ»»)"
2361| AOT_subst_fun_lambda0[AOT_substI]: "AOT_subst Θ ⟹ AOT_subst_fun (λ φ. (AOT_lambda0 (Θ φ)))"
2362| AOT_subst_fun_act[AOT_substI]: "AOT_subst Θ ⟹ AOT_subst_fun (λ φ. «𝒜«Θ φ»»)"
2363| AOT_subst_fun_box[AOT_substI]: "AOT_subst Θ ⟹ AOT_subst_fun (λ φ. «□«Θ φ»»)"
2364| AOT_subst_fun_def[AOT_substI]: "(⋀ φ . AOT_model_equiv_def (Θ φ) (Ψ φ)) ⟹ AOT_subst_fun Ψ ⟹ AOT_subst_fun Θ"
2365
2366instance proof
2367  fix ψ χ :: "'a ⇒ 'b" and φ :: ‹('a ⇒ 'b) ⇒ 𝗈›
2368  assume ‹AOT_subst φ›
2369  moreover assume cond: ‹AOT_subst_cond ψ χ›
2370  ultimately AOT_show ‹⊢□ «φ ψ» ≡ «φ χ»›
2371  proof(induct)
2372    case (AOT_subst_fun_const ψ)
2373    then show ?case by (simp add: "oth-class-taut:3:a")
2374  next
2375  case (AOT_subst_fun_id Ψ x)
2376  then show ?case by (simp add: AOT_subst AOT_subst_cond_fun_def) 
2377  next
2378  case (AOT_subst_fun_all Ψ Θ)
2379  AOT_have ‹⊢□ □(Θ{α, «ψ»} ≡ Θ{α, «χ»})› for α
2380    using AOT_subst_fun_all.hyps(3) AOT_subst_fun_all.prems RN by presburger
2381  thus ?case using AOT_subst[OF AOT_subst_fun_all(1)]
2382    by (simp add: RN "rule-sub-lem:1:d" AOT_subst_cond_fun_def AOT_subst_cond_𝗈_def)
2383  next
2384  case (AOT_subst_fun_not Ψ)
2385  then show ?case by (simp add: RN "rule-sub-lem:1:a")
2386  next
2387  case (AOT_subst_fun_imp Ψ Θ)
2388  then show ?case 
2389    unfolding AOT_subst_cond_fun_def AOT_subst_cond_𝗈_def
2390    by (meson "≡E"(5) "oth-class-taut:4:c" "oth-class-taut:4:d" "vdash-properties:6")
2391  next
2392  case (AOT_subst_fun_lambda0 Θ)
2393  then show ?case by (simp add: RN "rule-sub-lem:1:e")
2394  next
2395  case (AOT_subst_fun_act Θ)
2396  then show ?case by (simp add: RN "rule-sub-lem:1:f")
2397  next
2398  case (AOT_subst_fun_box Θ)
2399  then show ?case by (simp add: RN "rule-sub-lem:1:g")
2400  next
2401  case (AOT_subst_fun_def Θ Ψ)
2402  then show ?case
2403    by (meson "df-rules-formulas[3]" "df-rules-formulas[4]" "≡I" "≡E"(5))
2404  qed
2405qed
2406end
2407
2408method_setup AOT_defI =
2409‹Scan.lift (Scan.succeed (fn ctxt => (Method.CONTEXT_METHOD (fn thms => (Context_Tactic.CONTEXT_SUBGOAL (fn (trm,int) => 
2410Context_Tactic.CONTEXT_TACTIC (
2411let
2412fun findHeadConst (Const x) = SOME x
2413  | findHeadConst (A $ B) = findHeadConst A
2414  | findHeadConst _ = NONE
2415fun findDef (Const (const_name‹AOT_model_equiv_def›, _) $ lhs $ rhs) = findHeadConst lhs
2416  | findDef (A $ B) = (case findDef A of SOME x => SOME x | _ => findDef B)
2417  | findDef (Abs (a,b,c)) = findDef c
2418  | findDef _ = NONE
2419val const_opt = (findDef trm)
2420val defs = case const_opt of SOME const => List.filter (fn thm => let
2421    val concl = Thm.concl_of thm
2422    val thmconst = (findDef concl)
2423    in case thmconst of SOME (c,_) => fst const = c | _ => false end) (AOT_Definitions.get ctxt)
2424    | _ => []
2425in
2426resolve_tac ctxt defs 1
2427end
2428)) 1)))))›
2429‹Resolve AOT definitions›
2430
2431method AOT_subst_intro_helper = ((rule AOT_substI
2432      | AOT_defI
2433      | (simp only: AOT_subst_cond_𝗈_def AOT_subst_cond_fun_def; ((rule allI)+)?)))
2434
2435method AOT_subst for ψ::"'a::AOT_subst" and χ::"'a::AOT_subst" =
2436    (match conclusion in "[v ⊨ «φ ψ»]" for φ and v ⇒
2437      ‹match (φ) in "λa . ?p" ⇒ ‹fail› ¦ "λa . a" ⇒ ‹fail›
2438       ¦ _ ⇒ ‹rule AOT_subst[where φ=φ and ψ=ψ and χ=χ, THEN "≡E"(2)]
2439       ; (AOT_subst_intro_helper+)?››)
2440
2441method AOT_subst_rev for χ::"'a::AOT_subst" and ψ::"'a::AOT_subst" =
2442    (match conclusion in "[v ⊨ «φ ψ»]" for φ and v ⇒
2443      ‹match (φ) in "λa . ?p" ⇒ ‹fail› ¦ "λa . a" ⇒ ‹fail›
2444       ¦ _ ⇒ ‹rule AOT_subst[where φ=φ and ψ=χ and χ=ψ, THEN "≡E"(1)]
2445       ; (AOT_subst_intro_helper+)?››)
2446
2447method AOT_subst_manual for φ::"'a::AOT_subst ⇒ 𝗈" =
2448    (rule AOT_subst[where φ=φ, THEN "≡E"(2)]; (AOT_subst_intro_helper+)?)
2449
2450method AOT_subst_manual_rev for φ::"'a::AOT_subst ⇒ 𝗈" =
2451    (rule AOT_subst[where φ=φ, THEN "≡E"(1)]; (AOT_subst_intro_helper+)?)
2452
2453method AOT_subst_using uses subst =
2454    (match subst in "[?w ⊨ ψ ≡ χ]" for ψ χ ⇒ ‹
2455       match conclusion in "[v ⊨ «φ ψ»]" for φ v ⇒ ‹
2456         rule AOT_subst[where φ=φ and ψ=ψ and χ=χ, THEN "≡E"(2)]
2457         ; ((AOT_subst_intro_helper | (fact subst; fail))+)?››)
2458
2459method AOT_subst_using_rev uses subst =
2460    (match subst in "[?w ⊨ ψ ≡ χ]" for ψ χ ⇒ ‹
2461      match conclusion in "[v ⊨ «φ χ»]" for φ v ⇒ ‹
2462        rule AOT_subst[where φ=φ and ψ=ψ and χ=χ, THEN "≡E"(1)]
2463        ; ((AOT_subst_intro_helper | (fact subst; fail))+)?››)
2464
2465AOT_theorem "rule-sub-remark:1[1]": assumes ‹⊢□ A!x ≡ ¬◇E!x› and ‹¬A!x› shows ‹¬¬◇E!x›
2466  by (AOT_subst_rev "«A!x»" "«¬◇E!x»") (auto simp: assms)
2467
2468AOT_theorem "rule-sub-remark:1[2]": assumes ‹⊢□ A!x ≡ ¬◇E!x› and  ‹¬¬◇E!x› shows ‹¬A!x›
2469  by (AOT_subst "«A!x»" "«¬◇E!x»") (auto simp: assms)
2470
2471AOT_theorem "rule-sub-remark:2[1]":
2472  assumes ‹⊢□ [R]xy ≡ ([R]xy & ([Q]a ∨ ¬[Q]a))› and ‹p → [R]xy› shows ‹p → [R]xy & ([Q]a ∨ ¬[Q]a)›
2473  by (AOT_subst_using_rev subst: assms(1)) (simp add: assms(2))
2474
2475AOT_theorem "rule-sub-remark:2[2]":
2476  assumes ‹⊢□ [R]xy ≡ ([R]xy & ([Q]a ∨ ¬[Q]a))› and ‹p → [R]xy & ([Q]a ∨ ¬[Q]a)› shows ‹p → [R]xy›
2477  by (AOT_subst_using subst: assms(1)) (simp add: assms(2))
2478
2479AOT_theorem "rule-sub-remark:3[1]":
2480  assumes ‹for arbitrary x: ⊢□ A!x ≡ ¬◇E!x›
2481      and ‹∃x A!x›
2482    shows ‹∃x ¬◇E!x›
2483  by (AOT_subst_rev "λκ. «A!κ»" "λκ. «¬◇E!κ»") (auto simp: assms)
2484
2485AOT_theorem "rule-sub-remark:3[2]":
2486  assumes ‹for arbitrary x: ⊢□ A!x ≡ ¬◇E!x›
2487      and ‹∃x ¬◇E!x›
2488    shows ‹∃x A!x›
2489  by (AOT_subst "λκ. «A!κ»" "λκ. «¬◇E!κ»") (auto simp: assms)
2490
2491AOT_theorem "rule-sub-remark:4[1]":
2492  assumes ‹⊢□ ¬¬[P]x ≡ [P]x› and ‹𝒜¬¬[P]x› shows ‹𝒜[P]x›
2493  by (AOT_subst_using_rev subst: assms(1)) (simp add: assms(2))
2494
2495AOT_theorem "rule-sub-remark:4[2]":
2496  assumes ‹⊢□ ¬¬[P]x ≡ [P]x› and ‹𝒜[P]x› shows ‹𝒜¬¬[P]x›
2497  by (AOT_subst_using subst: assms(1)) (simp add: assms(2))
2498
2499AOT_theorem "rule-sub-remark:5[1]":
2500  assumes ‹⊢□ (φ → ψ) ≡ (¬ψ → ¬φ)› and ‹□(φ → ψ)› shows ‹□(¬ψ → ¬φ)›
2501  by (AOT_subst_using_rev subst: assms(1)) (simp add: assms(2))
2502
2503AOT_theorem "rule-sub-remark:5[2]":
2504  assumes ‹⊢□ (φ → ψ) ≡ (¬ψ → ¬φ)› and ‹□(¬ψ → ¬φ)› shows ‹□(φ → ψ)› 
2505  by (AOT_subst_using subst: assms(1)) (simp add: assms(2))
2506
2507AOT_theorem "rule-sub-remark:6[1]":
2508  assumes ‹⊢□ ψ ≡ χ› and ‹□(φ → ψ)› shows ‹□(φ → χ)› 
2509  by (AOT_subst_using_rev subst: assms(1)) (simp add: assms(2))
2510
2511AOT_theorem "rule-sub-remark:6[2]":
2512  assumes ‹⊢□ ψ ≡ χ› and ‹□(φ → χ)› shows ‹□(φ → ψ)›
2513  by (AOT_subst_using subst: assms(1)) (simp add: assms(2))
2514
2515AOT_theorem "rule-sub-remark:7[1]":
2516  assumes ‹⊢□ φ ≡ ¬¬φ› and ‹□(φ → φ)› shows ‹□(¬¬φ → φ)› 
2517  by (AOT_subst_using_rev subst: assms(1)) (simp add: assms(2))
2518
2519AOT_theorem "rule-sub-remark:7[2]":
2520  assumes ‹⊢□ φ ≡ ¬¬φ› and ‹□(¬¬φ → φ)› shows  ‹□(φ → φ)›
2521  by (AOT_subst_using subst: assms(1)) (simp add: assms(2))
2522
2523AOT_theorem "KBasic2:1": ‹□¬φ ≡ ¬◇φ›
2524  by (meson "conventions:5" "contraposition:2" "Hypothetical Syllogism" "df-rules-formulas[3]"
2525            "df-rules-formulas[4]" "≡I" "useful-tautologies:1")
2526
2527AOT_theorem "KBasic2:2": ‹◇(φ ∨ ψ) ≡ (◇φ ∨ ◇ψ)›
2528proof -
2529  AOT_have ‹◇(φ ∨ ψ) ≡ ◇¬(¬φ & ¬ψ)›
2530    by (simp add: "RE◇" "oth-class-taut:5:b")
2531  also AOT_have ‹… ≡ ¬□(¬φ & ¬ψ)›
2532    using "KBasic:11" "≡E"(6) "oth-class-taut:3:a" by blast
2533  also AOT_have ‹… ≡ ¬(□¬φ & □¬ψ)›
2534    using "KBasic:3" "≡E"(1) "oth-class-taut:4:b" by blast
2535  also AOT_have ‹… ≡ ¬(¬◇φ & ¬◇ψ)›
2536    apply (AOT_subst_rev "«□¬φ»" "«¬◇φ»")
2537    apply (simp add: "KBasic2:1")
2538    apply (AOT_subst_rev "«□¬ψ»" "«¬◇ψ»")
2539    by (auto simp: "KBasic2:1" "oth-class-taut:3:a")
2540  also AOT_have ‹… ≡ ¬¬(◇φ ∨ ◇ψ)›
2541    using "≡E"(6) "oth-class-taut:3:b" "oth-class-taut:5:b" by blast
2542  also AOT_have ‹… ≡ ◇φ ∨ ◇ψ›
2543    by (simp add: "≡I" "useful-tautologies:1" "useful-tautologies:2")
2544  finally show ?thesis .
2545qed
2546
2547AOT_theorem "KBasic2:3": ‹◇(φ & ψ) → (◇φ & ◇ψ)›
2548  by (metis "RM◇" "&I" "Conjunction Simplification"(1) "Conjunction Simplification"(2) "deduction-theorem" "modus-tollens:1" "reductio-aa:1")
2549
2550AOT_theorem "KBasic2:4": ‹◇(φ → ψ) ≡ (□φ → ◇ψ)›
2551proof -
2552  AOT_have ‹◇(φ → ψ) ≡ ◇(¬φ ∨ ψ)›
2553    by (AOT_subst "«φ → ψ»" "«¬φ ∨ ψ»")
2554       (auto simp: "oth-class-taut:1:c" "oth-class-taut:3:a")
2555  also AOT_have ‹... ≡ ◇¬φ ∨ ◇ψ›
2556    by (simp add: "KBasic2:2")
2557  also AOT_have ‹... ≡ ¬□φ ∨ ◇ψ›
2558    by (AOT_subst "«¬□φ»" "«◇¬φ»")
2559       (auto simp: "KBasic:11" "oth-class-taut:3:a")
2560  also AOT_have ‹... ≡ □φ → ◇ψ›
2561    using "≡E"(6) "oth-class-taut:1:c" "oth-class-taut:3:a" by blast
2562  finally show ?thesis .
2563qed
2564
2565AOT_theorem "KBasic2:5": ‹◇◇φ ≡ ¬□□¬φ›
2566  apply (AOT_subst "«◇φ»" "«¬□¬φ»")
2567   apply (simp add: "conventions:5" "≡Df")
2568  apply (AOT_subst "«◇¬□¬φ»" "«¬□¬¬□¬φ»")
2569   apply (simp add: "conventions:5" "≡Df")
2570  apply (AOT_subst_rev "«□¬φ»"  "«¬¬□¬φ»")
2571   apply (simp add: "oth-class-taut:3:b")
2572  by (simp add: "oth-class-taut:3:a")
2573
2574
2575AOT_theorem "KBasic2:6": ‹□(φ ∨ ψ) → (□φ ∨ ◇ψ)›
2576proof(rule "→I"; rule "raa-cor:1")
2577  AOT_assume ‹□(φ ∨ ψ)›
2578  AOT_hence ‹□(¬φ → ψ)›
2579    apply - apply (AOT_subst_rev "«φ ∨ ψ»" "«¬φ → ψ»")
2580    by (simp add: "conventions:2" "≡Df")
2581  AOT_hence 1: ‹◇¬φ → ◇ψ› using "KBasic:13" "vdash-properties:10" by blast
2582  AOT_assume ‹¬(□φ ∨ ◇ψ)›
2583  AOT_hence ‹¬□φ› and ‹¬◇ψ› using "&E" "≡E"(1) "oth-class-taut:5:d" by blast+
2584  AOT_thus ‹◇ψ & ¬◇ψ› using "&I"(1) 1[THEN "→E"] "KBasic:11" "≡E"(4) "raa-cor:3" by blast
2585qed
2586
2587AOT_theorem "KBasic2:7": ‹(□(φ ∨ ψ) & ◇¬φ) → ◇ψ›
2588proof(rule "→I"; frule "&E"(1); drule "&E"(2))
2589  AOT_assume ‹□(φ ∨ ψ)›
2590  AOT_hence 1: ‹□φ ∨ ◇ψ›
2591    using "KBasic2:6" "∨I"(2) "∨E"(1) by blast
2592  AOT_assume ‹◇¬φ›
2593  AOT_hence ‹¬□φ› using "KBasic:11" "≡E"(2) by blast
2594  AOT_thus ‹◇ψ› using 1 "∨E"(2) by blast
2595qed
2596
2597AOT_theorem "T-S5-fund:1": ‹φ → ◇φ›
2598  by (meson "≡dfI" "conventions:5" "contraposition:2" "Hypothetical Syllogism" "deduction-theorem" "qml:2"[axiom_inst])
2599lemmas "T◇" = "T-S5-fund:1"
2600
2601AOT_theorem "T-S5-fund:2": ‹◇□φ → □φ›
2602proof(rule "→I")
2603  AOT_assume ‹◇□φ›
2604  AOT_hence ‹¬□◇¬φ›
2605    using "KBasic:14" "≡E"(4) "raa-cor:3" by blast
2606  moreover AOT_have ‹◇¬φ → □◇¬φ›
2607    by (fact "qml:3"[axiom_inst])
2608  ultimately AOT_have ‹¬◇¬φ›
2609    using "modus-tollens:1" by blast
2610  AOT_thus ‹□φ› using "KBasic:12" "≡E"(2) by blast
2611qed
2612lemmas "5◇" = "T-S5-fund:2"
2613
2614(* Also interestingly none of these have proofs in PLM. *)
2615AOT_theorem "Act-Sub:1": ‹𝒜φ ≡ ¬𝒜¬φ›
2616  by (AOT_subst "«𝒜¬φ»" "«¬𝒜φ»")
2617     (auto simp: "logic-actual-nec:1"[axiom_inst] "oth-class-taut:3:b")
2618
2619AOT_theorem "Act-Sub:2": ‹◇φ ≡ 𝒜◇φ›
2620  apply (AOT_subst "«◇φ»" "«¬□¬φ»")
2621   apply (simp add: "conventions:5" "≡Df")
2622  by (metis "deduction-theorem" "≡I" "≡E"(1) "≡E"(2) "≡E"(3)
2623            "logic-actual-nec:1"[axiom_inst] "qml-act:2"[axiom_inst])
2624
2625AOT_theorem "Act-Sub:3": ‹𝒜φ → ◇φ›
2626  apply (AOT_subst "«◇φ»" "«¬□¬φ»")
2627   apply (simp add: "conventions:5" "≡Df")
2628  by (metis "Act-Sub:1" "deduction-theorem" "≡E"(4) "nec-imp-act" "reductio-aa:2" "vdash-properties:6")
2629
2630
2631AOT_theorem "Act-Sub:4": ‹𝒜φ ≡ ◇𝒜φ›
2632proof (rule "≡I"; rule "→I")
2633  AOT_assume ‹𝒜φ›
2634  AOT_thus ‹◇𝒜φ› using "T◇" "vdash-properties:10" by blast
2635next
2636  AOT_assume ‹◇𝒜φ›
2637  AOT_hence ‹¬□¬𝒜φ›
2638    using "≡dfE" "conventions:5" by blast
2639  AOT_hence ‹¬□𝒜¬φ›
2640    apply - apply (AOT_subst "«𝒜¬φ»" "«¬𝒜φ»")
2641    by (simp add: "logic-actual-nec:1"[axiom_inst])
2642  AOT_thus ‹𝒜φ›
2643      using "Act-Basic:1" "Act-Basic:6" "∨E"(3) "≡E"(4) "reductio-aa:1" by blast
2644qed
2645
2646AOT_theorem "Act-Sub:5": ‹◇𝒜φ → 𝒜◇φ›
2647  by (metis "Act-Sub:2" "Act-Sub:3" "Act-Sub:4" "deduction-theorem" "≡E"(1) "≡E"(2) "vdash-properties:6")
2648
2649AOT_theorem "S5Basic:1": ‹◇φ ≡ □◇φ›
2650  by (simp add: "≡I" "qml:2" "qml:3" "vdash-properties:1[2]")
2651
2652AOT_theorem "S5Basic:2": ‹□φ ≡ ◇□φ›
2653  by (simp add: "T◇" "5◇" "≡I")
2654
2655AOT_theorem "S5Basic:3": ‹φ → □◇φ›
2656  using "T◇" "Hypothetical Syllogism" "qml:3" "vdash-properties:1[2]" by blast
2657lemmas "B" = "S5Basic:3"
2658
2659AOT_theorem "S5Basic:4": ‹◇□φ → φ›
2660  using "5◇" "Hypothetical Syllogism" "qml:2" "vdash-properties:1[2]" by blast
2661lemmas "B◇" = "S5Basic:4"
2662
2663AOT_theorem "S5Basic:5": ‹□φ → □□φ›
2664  using "RM:1" "B" "5◇" "Hypothetical Syllogism" by blast
2665lemmas "4" = "S5Basic:5"
2666
2667AOT_theorem "S5Basic:6": ‹□φ ≡ □□φ›
2668  by (simp add: "4" "≡I" "qml:2"[axiom_inst])
2669
2670AOT_theorem "S5Basic:7": ‹◇◇φ → ◇φ›
2671  apply (AOT_subst "«◇◇φ»" "«¬□¬◇φ»")
2672   apply (simp add: "conventions:5" "≡Df")
2673  apply (AOT_subst "«◇φ»" "«¬□¬φ»")
2674   apply (simp add: "conventions:5" "≡Df")
2675  apply (AOT_subst_rev "«□¬φ»" "«¬¬□¬φ»")
2676   apply (simp add: "oth-class-taut:3:b")
2677  apply (AOT_subst_rev "«□¬φ»" "«□□¬φ»")
2678   apply (simp add: "S5Basic:6")
2679  by (simp add: "if-p-then-p")
2680
2681lemmas "4◇" = "S5Basic:7"
2682
2683AOT_theorem "S5Basic:8": ‹◇◇φ ≡ ◇φ›
2684  by (simp add: "4◇" "T◇" "≡I")
2685
2686AOT_theorem "S5Basic:9": ‹□(φ ∨ □ψ) ≡ (□φ ∨ □ψ)›
2687  apply (rule "≡I"; rule "→I")
2688  using "KBasic2:6" "5◇" "∨I"(3) "if-p-then-p" "vdash-properties:10" apply blast
2689  by (meson "KBasic:15" "4" "∨I"(3) "∨E"(1) "Disjunction Addition"(1) "con-dis-taut:7"
2690            "intro-elim:1" "Commutativity of ∨")
2691
2692AOT_theorem "S5Basic:10": ‹□(φ ∨ ◇ψ) ≡ (□φ ∨ ◇ψ)›
2693(* Note: nicely this proof is entirely sledgehammer generated *)
2694proof(rule "≡I"; rule "→I")
2695  AOT_assume ‹□(φ ∨ ◇ψ)›
2696  AOT_hence ‹□φ ∨ ◇◇ψ›
2697    by (meson "KBasic2:6" "∨I"(2) "∨E"(1))
2698  AOT_thus ‹□φ ∨ ◇ψ›
2699    by (meson "B◇" "4" "4◇" "T◇" "∨I"(3))
2700next
2701  AOT_assume ‹□φ ∨ ◇ψ›
2702  AOT_hence ‹□φ ∨ □◇ψ›
2703    by (meson "S5Basic:1" "B◇" "S5Basic:6" "T◇" "5◇" "∨I"(3) "intro-elim:1")
2704  AOT_thus ‹□(φ ∨ ◇ψ)›
2705    by (meson "KBasic:15" "∨I"(3) "∨E"(1) "Disjunction Addition"(1) "Disjunction Addition"(2))
2706qed
2707
2708AOT_theorem "S5Basic:11": ‹◇(φ & ◇ψ) ≡ (◇φ & ◇ψ)›
2709proof -
2710  AOT_have ‹◇(φ & ◇ψ) ≡ ◇¬(¬φ ∨ ¬◇ψ)›
2711    by (AOT_subst "«φ & ◇ψ»" "«¬(¬φ ∨ ¬◇ψ)»")
2712       (auto simp: "oth-class-taut:5:a" "oth-class-taut:3:a")
2713  also AOT_have ‹… ≡ ◇¬(¬φ ∨ □¬ψ)›
2714    by (AOT_subst "«□¬ψ»" "«¬◇ψ»")
2715       (auto simp: "KBasic2:1" "oth-class-taut:3:a")
2716  also AOT_have ‹… ≡ ¬□(¬φ ∨ □¬ψ)›
2717    using "KBasic:11" "≡E"(6) "oth-class-taut:3:a" by blast
2718  also AOT_have ‹… ≡ ¬(□¬φ ∨ □¬ψ)›
2719    using "S5Basic:9" "≡E"(1) "oth-class-taut:4:b" by blast
2720  also AOT_have ‹… ≡ ¬(¬◇φ ∨ ¬◇ψ)›
2721    apply (AOT_subst "«□¬φ»" "«¬◇φ»")
2722     apply (simp add: "KBasic2:1")
2723    apply (AOT_subst "«□¬ψ»" "«¬◇ψ»")
2724    by (auto simp: "KBasic2:1" "oth-class-taut:3:a")
2725  also AOT_have ‹… ≡ ◇φ & ◇ψ›
2726    using "≡E"(6) "oth-class-taut:3:a" "oth-class-taut:5:a" by blast
2727  finally show ?thesis .
2728qed
2729
2730AOT_theorem "S5Basic:12": ‹◇(φ & □ψ) ≡ (◇φ & □ψ)›
2731proof (rule "≡I"; rule "→I")
2732  AOT_assume ‹◇(φ & □ψ)›
2733  AOT_hence ‹◇φ & ◇□ψ›
2734    using "KBasic2:3" "vdash-properties:6" by blast
2735  AOT_thus ‹◇φ & □ψ›
2736    using "5◇" "&I" "&E"(1) "&E"(2) "vdash-properties:6" by blast
2737next
2738  AOT_assume ‹◇φ & □ψ›
2739  moreover AOT_have ‹(□□ψ & ◇φ) → ◇(φ & □ψ)›
2740    by (AOT_subst "«φ & □ψ»" "«□ψ & φ»")
2741       (auto simp: "Commutativity of &" "KBasic:16")
2742  ultimately AOT_show ‹◇(φ & □ψ)›
2743    by (metis "4" "&I" "Conjunction Simplification"(1) "Conjunction Simplification"(2) "vdash-properties:6")
2744qed
2745
2746
2747AOT_theorem "S5Basic:13": ‹□(φ → □ψ) ≡ □(◇φ → ψ)›
2748proof (rule "≡I")
2749  AOT_modally_strict {
2750    AOT_have ‹□(φ → □ψ) → (◇φ → ψ)›
2751      by (meson "KBasic:13" "B◇" "Hypothetical Syllogism" "deduction-theorem")
2752  }
2753  AOT_hence ‹□□(φ → □ψ) → □(◇φ → ψ)›
2754    by (rule RM)
2755  AOT_thus  ‹□(φ → □ψ) → □(◇φ → ψ)›
2756    using "4" "Hypothetical Syllogism" by blast
2757next
2758  AOT_modally_strict {
2759    AOT_have ‹□(◇φ → ψ) → (φ → □ψ)›
2760      by (meson "B" "Hypothetical Syllogism" "deduction-theorem" "qml:1" "vdash-properties:1[2]")
2761  }
2762  AOT_hence  ‹□□(◇φ → ψ) → □(φ → □ψ)›
2763    by (rule RM)
2764  AOT_thus ‹□(◇φ → ψ) → □(φ → □ψ)›
2765    using "4" "Hypothetical Syllogism" by blast
2766qed
2767
2768AOT_theorem "derived-S5-rules:1":
2769  assumes ‹Γ ⊢□ ◇φ → ψ› shows ‹□Γ ⊢□ φ → □ψ›
2770proof -
2771  AOT_have ‹□Γ ⊢□ □◇φ → □ψ›
2772    using assms by (rule "RM:1[prem]")
2773  AOT_thus ‹□Γ ⊢□ φ → □ψ›
2774    using "B" "Hypothetical Syllogism" by blast
2775qed
2776
2777AOT_theorem "derived-S5-rules:2":
2778  assumes ‹Γ ⊢□ φ → □ψ› shows ‹□Γ ⊢□ ◇φ → ψ›
2779proof -
2780  AOT_have ‹□Γ ⊢□ ◇φ → ◇□ψ›
2781    using assms by (rule "RM:2[prem]")
2782  AOT_thus ‹□Γ ⊢□ ◇φ → ψ›
2783    using "B◇" "Hypothetical Syllogism" by blast
2784qed
2785
2786AOT_theorem "BFs:1": ‹∀α □φ{α} → □∀α φ{α}›
2787proof -
2788  AOT_modally_strict {
2789    AOT_modally_strict {
2790      AOT_have ‹∀α □φ{α} → □φ{α}› for α by (fact AOT)
2791    }
2792    AOT_hence ‹◇∀α □φ{α} → ◇□φ{α}› for α by (rule "RM◇")
2793    AOT_hence ‹◇∀α □φ{α} → ∀α φ{α}›
2794      using "B◇" "∀I" "→E" "→I" by metis
2795  }
2796  thus ?thesis using "derived-S5-rules:1" by blast
2797qed
2798lemmas "BF" = "BFs:1"
2799
2800AOT_theorem "BFs:2": ‹□∀α φ{α} → ∀α □φ{α}›
2801proof -
2802  AOT_have ‹□∀α φ{α} → □φ{α}› for α using RM "cqt-orig:3" by metis
2803  thus ?thesis using  "cqt-orig:2"[THEN "→E"] "∀I" by metis
2804qed
2805lemmas "CBF" = "BFs:2"
2806
2807AOT_theorem "BFs:3": ‹◇∃α φ{α} → ∃α ◇φ{α}›
2808proof(rule "→I")
2809  AOT_modally_strict {
2810    AOT_have ‹□∀α ¬φ{α} ≡ ∀α □¬φ{α}›
2811      using BF CBF "≡I" by blast
2812  } note θ = this
2813
2814  AOT_assume ‹◇∃α φ{α}›
2815  AOT_hence ‹¬□¬(∃α φ{α})›
2816    using "≡dfE" "conventions:5" by blast
2817  AOT_hence ‹¬□∀α ¬φ{α}›
2818    apply - apply (AOT_subst "«∀α ¬φ{α}»" "«¬(∃α φ{α})»")
2819    using "≡dfI" "conventions:3" "conventions:4" "&I" "contraposition:2" "cqt-further:4"
2820          "df-rules-formulas[1]" "vdash-properties:1[2]" by blast
2821  AOT_hence ‹¬∀α □¬φ{α}›
2822    apply - apply (AOT_subst_using_rev subst: θ)
2823    using θ by blast
2824  AOT_hence ‹¬∀α ¬¬□¬φ{α}›
2825    apply - apply (AOT_subst_rev "λ τ. «□¬φ{τ}»"  "λ τ. «¬¬□¬φ{τ}»")
2826    by (simp add: "oth-class-taut:3:b")
2827  AOT_hence 0: ‹∃α ¬□¬φ{α}›
2828    by (rule "conventions:4"[THEN "≡dfI"])
2829  AOT_show ‹∃α ◇φ{α}›
2830    apply (AOT_subst "λ τ . «◇φ{τ}»" "λ τ . «¬□¬φ{τ}»")
2831     apply (simp add: "conventions:5" "≡Df")
2832    using 0 by blast
2833qed
2834lemmas "BF◇" = "BFs:3"
2835
2836AOT_theorem "BFs:4": ‹∃α ◇φ{α} → ◇∃α φ{α}›
2837proof(rule "→I")
2838  AOT_assume ‹∃α ◇φ{α}›
2839  AOT_hence ‹¬∀α ¬◇φ{α}›
2840    using "conventions:4"[THEN "≡dfE"] by blast
2841  AOT_hence ‹¬∀α □¬φ{α}›
2842    apply - apply (AOT_subst "λ τ . «□¬φ{τ}»" "λ τ . «¬◇φ{τ}»")
2843    by (simp add: "KBasic2:1")
2844  moreover AOT_have ‹∀α □¬φ{α} ≡ □∀α ¬φ{α}›
2845    using "≡I" "BF" "CBF" by metis
2846  ultimately AOT_have 1: ‹¬□∀α ¬φ{α}›
2847    using "≡E"(3) by blast
2848  AOT_show ‹◇∃α φ{α}›
2849    apply (rule "conventions:5"[THEN "≡dfI"])
2850    apply (AOT_subst "«∃α φ{α}»" "«¬∀α ¬φ{α}»")
2851     apply (simp add: "conventions:4" "≡Df")
2852    apply (AOT_subst "«¬¬∀α ¬φ{α}»" "«∀α ¬φ{α}»")
2853    by (auto simp: 1 "≡I" "useful-tautologies:1" "useful-tautologies:2")
2854qed
2855lemmas "CBF◇" = "BFs:4"
2856
2857AOT_theorem "sign-S5-thm:1": ‹∃α □φ{α} → □∃α φ{α}›
2858proof(rule "→I")
2859  AOT_assume ‹∃α □φ{α}›
2860  then AOT_obtain α where ‹□φ{α}› using "∃E" by metis
2861  moreover AOT_have ‹□α↓›
2862    by (simp add: "ex:1:a" "rule-ui:2[const_var]" RN)
2863  moreover AOT_have ‹□φ{τ}, □τ↓ ⊢□ □∃α φ{α}› for τ
2864  proof -
2865    AOT_have ‹φ{τ}, τ↓ ⊢□ ∃α φ{α}› using "existential:1" by blast
2866    AOT_thus ‹□φ{τ}, □τ↓ ⊢□ □∃α φ{α}›
2867      using "RN[prem]"[where Γ="{φ τ, «τ↓»}", simplified] by blast
2868  qed
2869  ultimately AOT_show ‹□∃α φ{α}› by blast
2870qed
2871lemmas Buridan = "sign-S5-thm:1"
2872
2873AOT_theorem "sign-S5-thm:2": ‹◇∀α φ{α} → ∀α ◇φ{α}›
2874proof -
2875  AOT_have ‹∀α (◇∀α φ{α} → ◇φ{α})›
2876    by (simp add: "RM◇" "cqt-orig:3" "∀I")
2877  AOT_thus ‹◇∀α φ{α} → ∀α ◇φ{α}›
2878    using "∀E"(4) "∀I" "→E" "→I" by metis
2879qed
2880lemmas "Buridan◇" = "sign-S5-thm:2"
2881
2882AOT_theorem "sign-S5-thm:3": ‹◇∃α (φ{α} & ψ{α}) → ◇(∃α φ{α} & ∃α ψ{α})›
2883  apply (rule "RM:2")
2884  by (metis (no_types, lifting) "instantiation" "&I" "&E"(1)
2885                                "&E"(2) "deduction-theorem" "existential:2[const_var]")
2886
2887AOT_theorem "sign-S5-thm:4": ‹◇∃α (φ{α} & ψ{α}) → ◇∃α φ{α}›
2888  apply (rule "RM:2")
2889  by (meson "instantiation" "&E"(1) "deduction-theorem" "existential:2[const_var]")
2890
2891AOT_theorem "sign-S5-thm:5": ‹(□∀α (φ{α} → ψ{α}) & □∀α (ψ{α} → χ{α})) → □∀α (φ{α} → χ{α})›
2892proof -
2893  {
2894    fix φ' ψ' χ'
2895    AOT_assume ‹⊢□ φ' & ψ' → χ'›
2896    AOT_hence ‹□φ' & □ψ' → □χ'›
2897      using "RN[prem]"[where Γ="{φ', ψ'}"] apply simp
2898      using "&E" "&I" "→E" "→I" by metis
2899  } note R = this
2900  show ?thesis by (rule R; fact AOT)
2901qed
2902
2903AOT_theorem "sign-S5-thm:6": ‹(□∀α (φ{α} ≡ ψ{α}) & □∀α(ψ{α} ≡ χ{α})) → □∀α(φ{α} ≡ χ{α})›
2904proof -
2905  {
2906    fix φ' ψ' χ'
2907    AOT_assume ‹⊢□ φ' & ψ' → χ'›
2908    AOT_hence ‹□φ' & □ψ' → □χ'›
2909      using "RN[prem]"[where Γ="{φ', ψ'}"] apply simp
2910      using "&E" "&I" "→E" "→I" by metis
2911  } note R = this
2912  show ?thesis by (rule R; fact AOT)
2913qed
2914
2915AOT_theorem "exist-nec2:1": ‹◇τ↓ → τ↓›
2916  using "B◇" "RM◇" "Hypothetical Syllogism" "exist-nec" by blast
2917
2918AOT_theorem "exists-nec2:2": ‹◇τ↓ ≡ □τ↓›
2919  by (meson "Act-Sub:3" "Hypothetical Syllogism" "exist-nec" "exist-nec2:1" "≡I" "nec-imp-act")
2920
2921AOT_theorem "exists-nec2:3": ‹¬τ↓ → □¬τ↓›
2922  using "KBasic2:1" "deduction-theorem" "exist-nec2:1" "≡E"(2) "modus-tollens:1" by blast
2923
2924AOT_theorem "exists-nec2:4": ‹◇¬τ↓ ≡ □¬τ↓›
2925  by (metis "Act-Sub:3" "KBasic:12" "deduction-theorem" "exist-nec" "exists-nec2:3" "≡I" "≡E"(4) "nec-imp-act" "reductio-aa:1")
2926
2927AOT_theorem "id-nec2:1": ‹◇α = β → α = β›
2928  using "B◇" "RM◇" "Hypothetical Syllogism" "id-nec:1" by blast
2929
2930AOT_theorem "id-nec2:2": ‹α ≠ β → □α ≠ β›
2931  apply (AOT_subst_using subst: "=-infix"[THEN "≡Df"])
2932  using "KBasic2:1" "deduction-theorem" "id-nec2:1" "≡E"(2) "modus-tollens:1" by blast
2933
2934AOT_theorem "id-nec2:3": ‹◇α ≠ β → α ≠ β›
2935  apply (AOT_subst_using subst: "=-infix"[THEN "≡Df"])
2936  by (metis "KBasic:11" "deduction-theorem" "id-nec:2" "≡E"(3) "reductio-aa:2" "vdash-properties:6")
2937
2938AOT_theorem "id-nec2:4": ‹◇α = β → □α = β›
2939  using "Hypothetical Syllogism" "id-nec2:1" "id-nec:1" by blast
2940
2941AOT_theorem "id-nec2:5": ‹◇α ≠ β → □α ≠ β›
2942  using "id-nec2:3" "id-nec2:2" "→I" "→E" by metis
2943
2944AOT_theorem "sc-eq-box-box:1": ‹□(φ → □φ) ≡ (◇φ → □φ)›
2945  apply (rule "≡I"; rule "→I")
2946  using "KBasic:13" "5◇" "Hypothetical Syllogism" "vdash-properties:10" apply blast
2947  by (metis "KBasic2:1" "KBasic:1" "KBasic:2" "S5Basic:13" "≡E"(2) "raa-cor:5" "vdash-properties:6")
2948
2949AOT_theorem "sc-eq-box-box:2": ‹(□(φ → □φ) ∨ (◇φ → □φ)) → (◇φ ≡ □φ)›
2950  by (metis "Act-Sub:3" "KBasic:13" "5◇" "∨E"(2) "deduction-theorem" "≡I" "nec-imp-act" "raa-cor:2" "vdash-properties:10")
2951
2952AOT_theorem "sc-eq-box-box:3": ‹□(φ → □φ) → (¬□φ ≡ □¬φ)›
2953proof (rule "→I"; rule "≡I"; rule "→I")
2954  AOT_assume ‹□(φ → □φ)›
2955  AOT_hence ‹◇φ → □φ› using "sc-eq-box-box:1" "≡E" by blast
2956  moreover AOT_assume ‹¬□φ›
2957  ultimately AOT_have ‹¬◇φ›
2958    using "modus-tollens:1" by blast
2959  AOT_thus ‹□¬φ›
2960    using "KBasic2:1" "≡E"(2) by blast
2961next
2962  AOT_assume ‹□(φ → □φ)›
2963  moreover AOT_assume ‹□¬φ›
2964  ultimately AOT_show ‹¬□φ›
2965    using "modus-tollens:1" "qml:2" "vdash-properties:10" "vdash-properties:1[2]" by blast
2966qed
2967
2968AOT_theorem "sc-eq-box-box:4": ‹(□(φ → □φ) & □(ψ → □ψ)) → ((□φ ≡ □ψ) → □(φ ≡ ψ))›
2969proof(rule "→I"; rule "→I")
2970  AOT_assume θ: ‹□(φ → □φ) & □(ψ → □ψ)›
2971  AOT_assume ξ: ‹□φ ≡ □ψ›
2972  AOT_hence ‹(□φ & □ψ) ∨ (¬□φ & ¬□ψ)›
2973    using "≡E"(4) "oth-class-taut:4:g" "raa-cor:3" by blast
2974  moreover {
2975    AOT_assume ‹□φ & □ψ›
2976    AOT_hence ‹□(φ ≡ ψ)›
2977      using "KBasic:3" "KBasic:8" "≡E"(2) "vdash-properties:10" by blast
2978  }
2979  moreover {
2980    AOT_assume ‹¬□φ & ¬□ψ›
2981    moreover AOT_have ‹¬□φ ≡ □¬φ› and ‹¬□ψ ≡ □¬ψ›
2982      using θ "Conjunction Simplification"(1) "Conjunction Simplification"(2) "sc-eq-box-box:3" "vdash-properties:10" by metis+
2983    ultimately AOT_have ‹□¬φ & □¬ψ›
2984      by (metis "&I" "Conjunction Simplification"(1) "Conjunction Simplification"(2) "≡E"(4) "modus-tollens:1" "raa-cor:3")
2985    AOT_hence ‹□(φ ≡ ψ)›
2986      using "KBasic:3" "KBasic:9" "≡E"(2) "vdash-properties:10" by blast
2987  }
2988  ultimately AOT_show ‹□(φ ≡ ψ)›
2989    using "∨E"(2) "reductio-aa:1" by blast
2990qed
2991
2992AOT_theorem "sc-eq-box-box:5": ‹(□(φ → □φ) & □(ψ → □ψ)) → □((φ ≡ ψ) → □(φ ≡ ψ))›
2993proof (rule "→I")
2994  AOT_assume ‹(□(φ → □φ) & □(ψ → □ψ))›
2995  AOT_hence ‹□(□(φ → □φ) & □(ψ → □ψ))›
2996    using 4[THEN "→E"] "&E" "&I" "KBasic:3" "≡E"(2) by metis
2997  moreover AOT_have ‹□(□(φ → □φ) & □(ψ → □ψ)) → □((φ ≡ ψ) → □(φ ≡ ψ))›
2998  proof (rule RM; rule "→I"; rule "→I")
2999    AOT_modally_strict {
3000      AOT_assume A: ‹(□(φ → □φ) & □(ψ → □ψ))›
3001      AOT_hence ‹φ → □φ› and ‹ψ → □ψ›
3002        using "&E" "qml:2"[axiom_inst] "→E" by blast+
3003      moreover AOT_assume ‹φ ≡ ψ›
3004      ultimately AOT_have ‹□φ ≡ □ψ›
3005        using "→E" "qml:2"[axiom_inst] "≡E" "≡I" by meson
3006      moreover AOT_have ‹(□φ ≡ □ψ) → □(φ ≡ ψ)›
3007        using A "sc-eq-box-box:4" "→E" by blast
3008      ultimately AOT_show ‹□(φ ≡ ψ)› using "→E" by blast
3009    }
3010  qed
3011  ultimately AOT_show ‹□((φ ≡ ψ) → □(φ ≡ ψ))› using "→E" by blast
3012qed
3013
3014AOT_theorem "sc-eq-box-box:6": ‹□(φ → □φ) → ((φ → □ψ) → □(φ → ψ))›
3015proof (rule "→I"; rule "→I"; rule "raa-cor:1")
3016  AOT_assume ‹¬□(φ → ψ)›
3017  AOT_hence 1: ‹◇¬(φ → ψ)› by (metis "KBasic:11" "≡E"(1))
3018  AOT_have ‹◇(φ & ¬ψ)›
3019    apply (AOT_subst ‹«φ & ¬ψ»› ‹«¬(φ → ψ)»›)
3020     apply (meson "Commutativity of ≡" "≡E"(1) "oth-class-taut:1:b")
3021    by (fact 1)
3022  AOT_hence ‹◇φ› and 2: ‹◇¬ψ› using "KBasic2:3"[THEN "→E"] "&E" by blast+
3023  moreover AOT_assume ‹□(φ → □φ)›
3024  ultimately AOT_have ‹□φ› by (metis "≡E"(1) "sc-eq-box-box:1" "→E")
3025  AOT_hence φ using "qml:2"[axiom_inst, THEN "→E"] by blast
3026  moreover AOT_assume ‹φ → □ψ›
3027  ultimately AOT_have ‹□ψ› using "→E" by blast
3028  moreover AOT_have ‹¬□ψ› using 2 "KBasic:12" "¬¬I" "intro-elim:3:d" by blast
3029  ultimately AOT_show ‹□ψ & ¬□ψ› using "&I" by blast
3030qed
3031
3032AOT_theorem "sc-eq-box-box:7": ‹□(φ → □φ) → ((φ → 𝒜ψ) → 𝒜(φ → ψ))›
3033proof (rule "→I"; rule "→I"; rule "raa-cor:1")
3034  AOT_assume ‹¬𝒜(φ → ψ)›
3035  AOT_hence 1: ‹𝒜¬(φ → ψ)› by (metis "Act-Basic:1" "∨E"(2))
3036  AOT_have ‹𝒜(φ & ¬ψ)›
3037    apply (AOT_subst ‹«φ & ¬ψ»› ‹«¬(φ → ψ)»›)
3038     apply (meson "Commutativity of ≡" "≡E"(1) "oth-class-taut:1:b")
3039    by (fact 1)
3040  AOT_hence ‹𝒜φ› and 2: ‹𝒜¬ψ› using "Act-Basic:2"[THEN "≡E"(1)] "&E" by blast+
3041  AOT_hence ‹◇φ› by (metis "Act-Sub:3" "→E")
3042  moreover AOT_assume ‹□(φ → □φ)›
3043  ultimately AOT_have ‹□φ› by (metis "≡E"(1) "sc-eq-box-box:1" "→E")
3044  AOT_hence φ using "qml:2"[axiom_inst, THEN "→E"] by blast
3045  moreover AOT_assume ‹φ → 𝒜ψ›
3046  ultimately AOT_have ‹𝒜ψ› using "→E" by blast
3047  moreover AOT_have ‹¬𝒜ψ› using 2 by (meson "Act-Sub:1" "≡E"(4) "raa-cor:3")
3048  ultimately AOT_show ‹𝒜ψ & ¬𝒜ψ› using "&I" by blast
3049qed
3050
3051AOT_theorem "sc-eq-fur:1": ‹◇𝒜φ ≡ □𝒜φ›
3052  using "Act-Basic:6" "Act-Sub:4" "≡E"(6) by blast
3053
3054AOT_theorem "sc-eq-fur:2": ‹□(φ → □φ) → (𝒜φ ≡ φ)›
3055  by (metis "B◇" "Act-Sub:3" "KBasic:13" "T◇" "Hypothetical Syllogism" "deduction-theorem" "≡I" "nec-imp-act")
3056
3057AOT_theorem "sc-eq-fur:3": ‹□∀x (φ{x} → □φ{x}) → (∃!x φ{x} → ιx φ{x}↓)›
3058proof (rule "→I"; rule "→I")
3059  AOT_assume ‹□∀x (φ{x} → □φ{x})›
3060  AOT_hence A: ‹∀x □(φ{x} → □φ{x})› using CBF "→E" by blast
3061  AOT_assume ‹∃!x φ{x}›
3062  then AOT_obtain a where a_def: ‹φ{a} & ∀y (φ{y} → y = a)›
3063    using "∃E"[rotated 1, OF "uniqueness:1"[THEN "≡dfE"]] by blast
3064  moreover AOT_have ‹□φ{a}› using calculation A "∀E"(2) "qml:2"[axiom_inst] "→E" "&E"(1) by blast
3065  AOT_hence ‹𝒜φ{a}› using "nec-imp-act" "vdash-properties:6" by blast
3066  moreover AOT_have ‹∀y (𝒜φ{y} → y = a)›
3067  proof (rule "∀I"; rule "→I")
3068    fix b
3069    AOT_assume ‹𝒜φ{b}›
3070    AOT_hence ‹◇φ{b}›
3071      using "Act-Sub:3" "vdash-properties:6" by blast
3072    moreover {
3073      AOT_have ‹□(φ{b} → □φ{b})›
3074        using A "∀E"(2) by blast
3075      AOT_hence ‹◇φ{b} → □φ{b}›
3076        using "KBasic:13" "5◇" "Hypothetical Syllogism" "vdash-properties:6" by blast
3077    }
3078    ultimately AOT_have ‹□φ{b}› using "→E" by blast
3079    AOT_hence ‹φ{b}› using "qml:2"[axiom_inst] "→E" by blast
3080    AOT_thus ‹b = a›
3081      using a_def[THEN "&E"(2)] "∀E"(2) "→E" by blast
3082  qed
3083  ultimately AOT_have ‹𝒜φ{a} & ∀y (𝒜φ{y} → y = a)›
3084    using "&I" by blast
3085  AOT_hence ‹∃x (𝒜φ{x} & ∀y (𝒜φ{y} → y = x))› using "∃I" by fast
3086  AOT_hence ‹∃!x 𝒜φ{x}› using "uniqueness:1"[THEN "≡dfI"] by fast
3087  AOT_thus ‹ιx φ{x}↓›
3088    using "actual-desc:1"[THEN "≡E"(2)] by blast
3089qed
3090
3091AOT_theorem "sc-eq-fur:4": ‹□∀x (φ{x} → □φ{x}) → (x = ιx φ{x} ≡ (φ{x} & ∀z (φ{z} → z = x)))›
3092proof (rule "→I")
3093  AOT_assume ‹□∀x (φ{x} → □φ{x})›
3094  AOT_hence ‹∀x □(φ{x} → □φ{x})› using CBF "→E" by blast
3095  AOT_hence A: ‹𝒜φ{α} ≡ φ{α}› for α using "sc-eq-fur:2" "∀E" "→E" by fast
3096  AOT_show ‹x = ιx φ{x} ≡ (φ{x} & ∀z (φ{z} → z = x))›
3097  proof (rule "≡I"; rule "→I")
3098    AOT_assume ‹x = ιx φ{x}›
3099    AOT_hence B: ‹𝒜φ{x} & ∀z (𝒜φ{z} → z = x)›
3100      using "nec-hintikka-scheme"[THEN "≡E"(1)] by blast
3101    AOT_show ‹φ{x} & ∀z (φ{z} → z = x)›
3102    proof (rule "&I"; (rule "∀I"; rule "→I")?)
3103      AOT_show ‹φ{x}› using A B[THEN "&E"(1)] "≡E"(1) by blast
3104    next
3105      AOT_show ‹z = x› if ‹φ{z}› for z
3106        using that B[THEN "&E"(2)] "∀E"(2) "→E" A[THEN "≡E"(2)] by blast
3107    qed
3108  next
3109    AOT_assume B: ‹φ{x} & ∀z (φ{z} → z = x)›
3110    AOT_have ‹𝒜φ{x} & ∀z (𝒜φ{z} → z = x)›
3111    proof(rule "&I"; (rule "∀I"; rule "→I")?)
3112      AOT_show ‹𝒜φ{x}› using B[THEN "&E"(1)] A[THEN "≡E"(2)] by blast
3113    next
3114      AOT_show ‹b = x› if ‹𝒜φ{b}› for b
3115        using that A[THEN "≡E"(1)] B[THEN "&E"(2), THEN "∀E"(2), THEN "→E"] by blast
3116    qed
3117    AOT_thus ‹x = ιx φ{x}›
3118      using "nec-hintikka-scheme"[THEN "≡E"(2)] by blast
3119  qed
3120qed
3121
3122AOT_theorem "id-act:1": ‹α = β ≡ 𝒜α = β›
3123  by (meson "Act-Sub:3" "Hypothetical Syllogism" "id-nec2:1" "id-nec:2" "≡I" "nec-imp-act")
3124
3125AOT_theorem "id-act:2": ‹α ≠ β ≡ 𝒜α ≠ β›
3126proof (AOT_subst "«α ≠ β»" "«¬(α = β)»")
3127  AOT_modally_strict {
3128    AOT_show ‹α ≠ β ≡ ¬(α = β)›
3129      by (simp add: "=-infix" "≡Df")
3130  }
3131next
3132  AOT_show ‹¬(α = β) ≡ 𝒜¬(α = β)›
3133  proof (safe intro!: "≡I" "→I")
3134    AOT_assume ‹¬α = β›
3135    AOT_hence ‹¬𝒜α = β› using "id-act:1" "≡E"(3) by blast
3136    AOT_thus ‹𝒜¬α = β›
3137      using "¬¬E" "Act-Sub:1" "≡E"(3) by blast
3138  next
3139    AOT_assume ‹𝒜¬α = β›
3140    AOT_hence ‹¬𝒜α = β›
3141      using "¬¬I" "Act-Sub:1" "≡E"(4) by blast
3142    AOT_thus ‹¬α = β›
3143      using "id-act:1" "≡E"(4) by blast
3144  qed
3145qed
3146
3147AOT_theorem "A-Exists:1": ‹𝒜∃!α φ{α} ≡ ∃!α 𝒜φ{α}›
3148proof -
3149  AOT_have ‹𝒜∃!α φ{α} ≡ 𝒜∃α∀β (φ{β} ≡ β = α)›
3150    by (AOT_subst_using subst: "uniqueness:2")
3151       (simp add: "oth-class-taut:3:a")
3152  also AOT_have ‹… ≡ ∃α 𝒜∀β (φ{β} ≡ β = α)›
3153    by (simp add: "Act-Basic:10")
3154  also AOT_have ‹… ≡ ∃α∀β 𝒜(φ{β} ≡ β = α)›
3155    by (AOT_subst "λ τ . «𝒜∀β (φ{β} ≡ β = τ)»" "λ τ . «∀β 𝒜(φ{β} ≡ β = τ)»")
3156       (auto simp: "logic-actual-nec:3" "vdash-properties:1[2]" "oth-class-taut:3:a")
3157  also AOT_have ‹… ≡ ∃α∀β (𝒜φ{β} ≡ 𝒜β = α)›
3158    by (AOT_subst_rev "λ τ τ' . «𝒜(φ{τ'} ≡ τ' = τ)»" "λ τ τ'. «𝒜φ{τ'} ≡ 𝒜τ' = τ»")
3159       (auto simp: "Act-Basic:5" "cqt-further:7")
3160  also AOT_have ‹… ≡ ∃α∀β (𝒜φ{β} ≡ β = α)›
3161    apply (AOT_subst "λ τ τ' :: 'a . «𝒜τ' = τ»" "λ τ τ'. «τ' = τ»")
3162     apply (meson "id-act:1" "≡E"(6) "oth-class-taut:3:a")
3163    by (simp add: "cqt-further:7")
3164  also AOT_have ‹... ≡ ∃!α 𝒜φ{α}›
3165    using "uniqueness:2" "Commutativity of ≡"[THEN "≡E"(1)] by fast
3166  finally show ?thesis .
3167qed
3168
3169AOT_theorem "A-Exists:2": ‹ιx φ{x}↓ ≡ 𝒜∃!x φ{x}›
3170  by (AOT_subst_using subst: "A-Exists:1")
3171     (simp add: "actual-desc:1")
3172
3173AOT_theorem "id-act-desc:1": ‹ιx (x = y)↓›
3174proof(rule "existence:1"[THEN "≡dfI"]; rule "∃I")
3175  AOT_show ‹[λx E!x → E!x]ιx (x = y)›
3176  proof (rule "russell-axiom[exe,1].nec-russell-axiom"[THEN "≡E"(2)]; rule "∃I"; (rule "&I")+)
3177    AOT_show ‹𝒜y = y› by (simp add: "RA[2]" "id-eq:1")
3178  next
3179    AOT_show ‹∀z (𝒜z = y → z = y)›
3180      apply (rule "∀I")
3181      using "id-act:1"[THEN "≡E"(2)] "→I" by blast
3182  next
3183    AOT_show ‹[λx E!x → E!x]y›
3184    proof (rule "lambda-predicates:2"[axiom_inst, THEN "→E", THEN "≡E"(2)])
3185      AOT_show ‹[λx E!x → E!x]↓›
3186        by "cqt:2[lambda]"
3187    next
3188      AOT_show ‹E!y → E!y› 
3189        by (simp add: "if-p-then-p")
3190    qed
3191  qed
3192next
3193  AOT_show ‹[λx E!x → E!x]↓›
3194    by "cqt:2[lambda]"
3195qed
3196
3197AOT_theorem "id-act-desc:2": ‹y = ιx (x = y)›
3198  by (rule descriptions[axiom_inst, THEN "≡E"(2)]; rule "∀I"; rule "id-act:1"[symmetric])
3199
3200AOT_theorem "pre-en-eq:1[1]": ‹x1[F] → □x1[F]›
3201  by (simp add: encoding "vdash-properties:1[2]")
3202
3203AOT_theorem "pre-en-eq:1[2]": ‹x1x2[F] → □x1x2[F]›
3204proof (rule "→I")
3205  AOT_assume ‹x1x2[F]›
3206  AOT_hence ‹x1[λy [F]yx2]› and ‹x2[λy [F]x1y]›
3207    using "nary-encoding[2]"[axiom_inst, THEN "≡E"(1)] "&E" by blast+
3208  moreover AOT_have ‹[λy [F]yx2]↓› by "cqt:2[lambda]"
3209  moreover AOT_have ‹[λy [F]x1y]↓› by "cqt:2[lambda]"
3210  ultimately AOT_have ‹□x1[λy [F]yx2]› and ‹□x2[λy [F]x1y]›
3211    using encoding[axiom_inst, unvarify F] "→E" "&I" by blast+
3212  note A = this
3213  AOT_hence ‹□(x1[λy [F]yx2] & x2[λy [F]x1y])›
3214    using "KBasic:3"[THEN "≡E"(2)] "&I" by blast
3215  AOT_thus ‹□x1x2[F]›
3216    by (rule "nary-encoding[2]"[axiom_inst, THEN RN, THEN "KBasic:6"[THEN "→E"], THEN "≡E"(2)])
3217qed
3218
3219AOT_theorem "pre-en-eq:1[3]": ‹x1x2x3[F] → □x1x2x3[F]›
3220proof (rule "→I")
3221  AOT_assume ‹x1x2x3[F]›
3222  AOT_hence ‹x1[λy [F]yx2x3]› and ‹x2[λy [F]x1yx3]› and ‹x3[λy [F]x1x2y]›
3223    using "nary-encoding[3]"[axiom_inst, THEN "≡E"(1)] "&E" by blast+
3224  moreover AOT_have ‹[λy [F]yx2x3]↓› by "cqt:2[lambda]"
3225  moreover AOT_have ‹[λy [F]x1yx3]↓› by "cqt:2[lambda]"
3226  moreover AOT_have ‹[λy [F]x1x2y]↓› by "cqt:2[lambda]"
3227  ultimately AOT_have ‹□x1[λy [F]yx2x3]› and ‹□x2[λy [F]x1yx3]› and ‹□x3[λy [F]x1x2y]›
3228    using encoding[axiom_inst, unvarify F] "→E" by blast+
3229  note A = this
3230  AOT_have B: ‹□(x1[λy [F]yx2x3] & x2[λy [F]x1yx3] & x3[λy [F]x1x2y])›
3231    by (rule "KBasic:3"[THEN "≡E"(2)] "&I" A)+
3232  AOT_thus ‹□x1x2x3[F]›
3233    by (rule "nary-encoding[3]"[axiom_inst, THEN RN, THEN "KBasic:6"[THEN "→E"], THEN "≡E"(2)])
3234qed
3235
3236AOT_theorem "pre-en-eq:1[4]": ‹x1x2x3x4[F] → □x1x2x3x4[F]›
3237proof (rule "→I")
3238  AOT_assume ‹x1x2x3x4[F]›
3239  AOT_hence ‹x1[λy [F]yx2x3x4]› and ‹x2[λy [F]x1yx3x4]› and ‹x3[λy [F]x1x2yx4]› and  ‹x4[λy [F]x1x2x3y]›
3240    using "nary-encoding[4]"[axiom_inst, THEN "≡E"(1)] "&E" by metis+
3241  moreover AOT_have ‹[λy [F]yx2x3x4]↓› by "cqt:2[lambda]"
3242  moreover AOT_have ‹[λy [F]x1yx3x4]↓› by "cqt:2[lambda]"
3243  moreover AOT_have ‹[λy [F]x1x2yx4]↓› by "cqt:2[lambda]"
3244  moreover AOT_have ‹[λy [F]x1x2x3y]↓› by "cqt:2[lambda]"
3245  ultimately AOT_have ‹□x1[λy [F]yx2x3x4]› and ‹□x2[λy [F]x1yx3x4]› and ‹□x3[λy [F]x1x2yx4]› and ‹□x4[λy [F]x1x2x3y]›
3246    using "→E" encoding[axiom_inst, unvarify F] by blast+
3247  note A = this
3248  AOT_have B: ‹□(x1[λy [F]yx2x3x4] & x2[λy [F]x1yx3x4] & x3[λy [F]x1x2yx4] & x4[λy [F]x1x2x3y])›
3249    by (rule "KBasic:3"[THEN "≡E"(2)] "&I" A)+
3250  AOT_thus ‹□x1x2x3x4[F]›
3251    by (rule "nary-encoding[4]"[axiom_inst, THEN RN, THEN "KBasic:6"[THEN "→E"], THEN "≡E"(2)])
3252qed
3253
3254AOT_theorem "pre-en-eq:2[1]": ‹¬x1[F] → □¬x1[F]›
3255proof (rule "→I"; rule "raa-cor:1")
3256  AOT_assume ‹¬□¬x1[F]›
3257  AOT_hence ‹◇x1[F]›
3258    by (rule "conventions:5"[THEN "≡dfI"])
3259  AOT_hence ‹x1[F]›
3260    by(rule "S5Basic:13"[THEN "≡E"(1), OF  "pre-en-eq:1[1]"[THEN RN], THEN "qml:2"[axiom_inst, THEN "→E"], THEN "→E"])
3261  moreover AOT_assume ‹¬x1[F]›
3262  ultimately AOT_show ‹x1[F] & ¬x1[F]› by (rule "&I")
3263qed
3264AOT_theorem "pre-en-eq:2[2]": ‹¬x1x2[F] → □¬x1x2[F]›
3265proof (rule "→I"; rule "raa-cor:1")
3266  AOT_assume ‹¬□¬x1x2[F]›
3267  AOT_hence ‹◇x1x2[F]›
3268    by (rule "conventions:5"[THEN "≡dfI"])
3269  AOT_hence ‹x1x2[F]›
3270    by(rule "S5Basic:13"[THEN "≡E"(1), OF  "pre-en-eq:1[2]"[THEN RN], THEN "qml:2"[axiom_inst, THEN "→E"], THEN "→E"])
3271  moreover AOT_assume ‹¬x1x2[F]›
3272  ultimately AOT_show ‹x1x2[F] & ¬x1x2[F]› by (rule "&I")
3273qed
3274
3275AOT_theorem "pre-en-eq:2[3]": ‹¬x1x2x3[F] → □¬x1x2x3[F]›
3276proof (rule "→I"; rule "raa-cor:1")
3277  AOT_assume ‹¬□¬x1x2x3[F]›
3278  AOT_hence ‹◇x1x2x3[F]›
3279    by (rule "conventions:5"[THEN "≡dfI"])
3280  AOT_hence ‹x1x2x3[F]›
3281    by(rule "S5Basic:13"[THEN "≡E"(1), OF  "pre-en-eq:1[3]"[THEN RN], THEN "qml:2"[axiom_inst, THEN "→E"], THEN "→E"])
3282  moreover AOT_assume ‹¬x1x2x3[F]›
3283  ultimately AOT_show ‹x1x2x3[F] & ¬x1x2x3[F]› by (rule "&I")
3284qed
3285
3286AOT_theorem "pre-en-eq:2[4]": ‹¬x1x2x3x4[F] → □¬x1x2x3x4[F]›
3287proof (rule "→I"; rule "raa-cor:1")
3288  AOT_assume ‹¬□¬x1x2x3x4[F]›
3289  AOT_hence ‹◇x1x2x3x4[F]›
3290    by (rule "conventions:5"[THEN "≡dfI"])
3291  AOT_hence ‹x1x2x3x4[F]›
3292    by(rule "S5Basic:13"[THEN "≡E"(1), OF  "pre-en-eq:1[4]"[THEN RN], THEN "qml:2"[axiom_inst, THEN "→E"], THEN "→E"])
3293  moreover AOT_assume ‹¬x1x2x3x4[F]›
3294  ultimately AOT_show ‹x1x2x3x4[F] & ¬x1x2x3x4[F]› by (rule "&I")
3295qed
3296
3297AOT_theorem "en-eq:1[1]": ‹◇x1[F] ≡ □x1[F]›
3298  using "pre-en-eq:1[1]"[THEN RN] "sc-eq-box-box:2" "∨I" "→E" by metis
3299AOT_theorem "en-eq:1[2]": ‹◇x1x2[F] ≡ □x1x2[F]›
3300  using "pre-en-eq:1[2]"[THEN RN] "sc-eq-box-box:2" "∨I" "→E" by metis
3301AOT_theorem "en-eq:1[3]": ‹◇x1x2x3[F] ≡ □x1x2x3[F]›
3302  using "pre-en-eq:1[3]"[THEN RN] "sc-eq-box-box:2" "∨I" "→E" by fast
3303AOT_theorem "en-eq:1[4]": ‹◇x1x2x3x4[F] ≡ □x1x2x3x4[F]›
3304  using "pre-en-eq:1[4]"[THEN RN] "sc-eq-box-box:2" "∨I" "→E" by fast
3305
3306AOT_theorem "en-eq:2[1]": ‹x1[F] ≡ □x1[F]›
3307  by (simp add: "≡I" "pre-en-eq:1[1]" "qml:2"[axiom_inst])
3308AOT_theorem "en-eq:2[2]": ‹x1x2[F] ≡ □x1x2[F]›
3309  by (simp add: "≡I" "pre-en-eq:1[2]" "qml:2"[axiom_inst])
3310AOT_theorem "en-eq:2[3]": ‹x1x2x3[F] ≡ □x1x2x3[F]›
3311  by (simp add: "≡I" "pre-en-eq:1[3]" "qml:2"[axiom_inst])
3312AOT_theorem "en-eq:2[4]": ‹x1x2x3x4[F] ≡ □x1x2x3x4[F]›
3313  by (simp add: "≡I" "pre-en-eq:1[4]" "qml:2"[axiom_inst])
3314
3315AOT_theorem "en-eq:3[1]": ‹◇x1[F] ≡ x1[F]›
3316  using "T◇" "derived-S5-rules:2"[where Γ="{}", OF "pre-en-eq:1[1]"] "≡I" by blast
3317AOT_theorem "en-eq:3[2]": ‹◇x1x2[F] ≡ x1x2[F]›
3318  using "T◇" "derived-S5-rules:2"[where Γ="{}", OF "pre-en-eq:1[2]"] "≡I" by blast
3319AOT_theorem "en-eq:3[3]": ‹◇x1x2x3[F] ≡ x1x2x3[F]›
3320  using "T◇" "derived-S5-rules:2"[where Γ="{}", OF "pre-en-eq:1[3]"] "≡I" by blast
3321AOT_theorem "en-eq:3[4]": ‹◇x1x2x3x4[F] ≡ x1x2x3x4[F]›
3322  using "T◇" "derived-S5-rules:2"[where Γ="{}", OF "pre-en-eq:1[4]"] "≡I" by blast
3323
3324AOT_theorem "en-eq:4[1]": ‹(x1[F] ≡ y1[G]) ≡ (□x1[F] ≡ □y1[G])›
3325  apply (rule "≡I"; rule "→I"; rule "≡I"; rule "→I")
3326  using "qml:2"[axiom_inst, THEN "→E"] "≡E"(1,2) "en-eq:2[1]" by blast+
3327AOT_theorem "en-eq:4[2]": ‹(x1x2[F] ≡ y1y2[G]) ≡ (□x1x2[F] ≡ □y1y2[G])›
3328  apply (rule "≡I"; rule "→I"; rule "≡I"; rule "→I")
3329  using "qml:2"[axiom_inst, THEN "→E"] "≡E"(1,2) "en-eq:2[2]" by blast+
3330AOT_theorem "en-eq:4[3]": ‹(x1x2x3[F] ≡ y1y2y3[G]) ≡ (□x1x2x3[F] ≡ □y1y2y3[G])›
3331  apply (rule "≡I"; rule "→I"; rule "≡I"; rule "→I")
3332  using "qml:2"[axiom_inst, THEN "→E"] "≡E"(1,2) "en-eq:2[3]" by blast+
3333AOT_theorem "en-eq:4[4]": ‹(x1x2x3x4[F] ≡ y1y2y3y4[G]) ≡ (□x1x2x3x4[F] ≡ □y1y2y3y4[G])›
3334  apply (rule "≡I"; rule "→I"; rule "≡I"; rule "→I")
3335  using "qml:2"[axiom_inst, THEN "→E"] "≡E"(1,2) "en-eq:2[4]" by blast+
3336
3337AOT_theorem "en-eq:5[1]": ‹□(x1[F] ≡ y1[G]) ≡ (□x1[F] ≡ □y1[G])›
3338  apply (rule "≡I"; rule "→I")
3339  using "en-eq:4[1]"[THEN "≡E"(1)] "qml:2"[axiom_inst, THEN "→E"] apply blast
3340  using "sc-eq-box-box:4"[THEN "→E", THEN "→E"]
3341        "&I"[OF "pre-en-eq:1[1]"[THEN RN], OF "pre-en-eq:1[1]"[THEN RN]] by blast
3342AOT_theorem "en-eq:5[2]": ‹□(x1x2[F] ≡ y1y2[G]) ≡ (□x1x2[F] ≡ □y1y2[G])›
3343  apply (rule "≡I"; rule "→I")
3344  using "en-eq:4[2]"[THEN "≡E"(1)] "qml:2"[axiom_inst, THEN "→E"] apply blast
3345  using "sc-eq-box-box:4"[THEN "→E", THEN "→E"]
3346        "&I"[OF "pre-en-eq:1[2]"[THEN RN], OF "pre-en-eq:1[2]"[THEN RN]] by blast
3347AOT_theorem "en-eq:5[3]": ‹□(x1x2x3[F] ≡ y1y2y3[G]) ≡ (□x1x2x3[F] ≡ □y1y2y3[G])›
3348  apply (rule "≡I"; rule "→I")
3349  using "en-eq:4[3]"[THEN "≡E"(1)] "qml:2"[axiom_inst, THEN "→E"] apply blast
3350  using "sc-eq-box-box:4"[THEN "→E", THEN "→E"]
3351        "&I"[OF "pre-en-eq:1[3]"[THEN RN], OF "pre-en-eq:1[3]"[THEN RN]] by blast
3352AOT_theorem "en-eq:5[4]": ‹□(x1x2x3x4[F] ≡ y1y2y3y4[G]) ≡ (□x1x2x3x4[F] ≡ □y1y2y3y4[G])›
3353  apply (rule "≡I"; rule "→I")
3354  using "en-eq:4[4]"[THEN "≡E"(1)] "qml:2"[axiom_inst, THEN "→E"] apply blast
3355  using "sc-eq-box-box:4"[THEN "→E", THEN "→E"]
3356        "&I"[OF "pre-en-eq:1[4]"[THEN RN], OF "pre-en-eq:1[4]"[THEN RN]] by blast
3357
3358AOT_theorem "en-eq:6[1]": ‹(x1[F] ≡ y1[G]) ≡ □(x1[F] ≡ y1[G])›
3359  using "en-eq:5[1]"[symmetric] "en-eq:4[1]" "≡E"(5) by fast
3360AOT_theorem "en-eq:6[2]": ‹(x1x2[F] ≡ y1y2[G]) ≡ □(x1x2[F] ≡ y1y2[G])›
3361  using "en-eq:5[2]"[symmetric] "en-eq:4[2]" "≡E"(5) by fast
3362AOT_theorem "en-eq:6[3]": ‹(x1x2x3[F] ≡ y1y2y3[G]) ≡ □(x1x2x3[F] ≡ y1y2y3[G])›
3363  using "en-eq:5[3]"[symmetric] "en-eq:4[3]" "≡E"(5) by fast
3364AOT_theorem "en-eq:6[4]": ‹(x1x2x3x4[F] ≡ y1y2y3y4[G]) ≡ □(x1x2x3x4[F] ≡ y1y2y3y4[G])›
3365  using "en-eq:5[4]"[symmetric] "en-eq:4[4]" "≡E"(5) by fast
3366
3367AOT_theorem "en-eq:7[1]": ‹¬x1[F] ≡ □¬x1[F]›
3368  using "pre-en-eq:2[1]" "qml:2"[axiom_inst] "≡I" by blast
3369AOT_theorem "en-eq:7[2]": ‹¬x1x2[F] ≡ □¬x1x2[F]›
3370  using "pre-en-eq:2[2]" "qml:2"[axiom_inst] "≡I" by blast
3371AOT_theorem "en-eq:7[3]": ‹¬x1x2x3[F] ≡ □¬x1x2x3[F]›
3372  using "pre-en-eq:2[3]" "qml:2"[axiom_inst] "≡I" by blast
3373AOT_theorem "en-eq:7[4]": ‹¬x1x2x3x4[F] ≡ □¬x1x2x3x4[F]›
3374  using "pre-en-eq:2[4]" "qml:2"[axiom_inst] "≡I" by blast
3375
3376AOT_theorem "en-eq:8[1]": ‹◇¬x1[F] ≡ ¬x1[F]›
3377  using "en-eq:2[1]"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)]] "KBasic:11" "≡E"(5)[symmetric] by blast
3378AOT_theorem "en-eq:8[2]": ‹◇¬x1x2[F] ≡ ¬x1x2[F]›
3379  using "en-eq:2[2]"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)]] "KBasic:11" "≡E"(5)[symmetric] by blast
3380AOT_theorem "en-eq:8[3]": ‹◇¬x1x2x3[F] ≡ ¬x1x2x3[F]›
3381  using "en-eq:2[3]"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)]] "KBasic:11" "≡E"(5)[symmetric] by blast
3382AOT_theorem "en-eq:8[4]": ‹◇¬x1x2x3x4[F] ≡ ¬x1x2x3x4[F]›
3383  using "en-eq:2[4]"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)]] "KBasic:11" "≡E"(5)[symmetric] by blast
3384
3385AOT_theorem "en-eq:9[1]": ‹◇¬x1[F] ≡ □¬x1[F]›
3386  using "en-eq:7[1]" "en-eq:8[1]" "≡E"(5) by blast
3387AOT_theorem "en-eq:9[2]": ‹◇¬x1x2[F] ≡ □¬x1x2[F]›
3388  using "en-eq:7[2]" "en-eq:8[2]" "≡E"(5) by blast
3389AOT_theorem "en-eq:9[3]": ‹◇¬x1x2x3[F] ≡ □¬x1x2x3[F]›
3390  using "en-eq:7[3]" "en-eq:8[3]" "≡E"(5) by blast
3391AOT_theorem "en-eq:9[4]": ‹◇¬x1x2x3x4[F] ≡ □¬x1x2x3x4[F]›
3392  using "en-eq:7[4]" "en-eq:8[4]" "≡E"(5) by blast
3393
3394AOT_theorem "en-eq:10[1]": ‹𝒜x1[F] ≡ x1[F]›
3395  by (metis "Act-Sub:3" "deduction-theorem" "≡I" "≡E"(1) "nec-imp-act" "en-eq:3[1]" "pre-en-eq:1[1]")
3396AOT_theorem "en-eq:10[2]": ‹𝒜x1x2[F] ≡ x1x2[F]›
3397  by (metis "Act-Sub:3" "deduction-theorem" "≡I" "≡E"(1) "nec-imp-act" "en-eq:3[2]" "pre-en-eq:1[2]")
3398AOT_theorem "en-eq:10[3]": ‹𝒜x1x2x3[F] ≡ x1x2x3[F]›
3399  by (metis "Act-Sub:3" "deduction-theorem" "≡I" "≡E"(1) "nec-imp-act" "en-eq:3[3]" "pre-en-eq:1[3]")
3400AOT_theorem "en-eq:10[4]": ‹𝒜x1x2x3x4[F] ≡ x1x2x3x4[F]›
3401  by (metis "Act-Sub:3" "deduction-theorem" "≡I" "≡E"(1) "nec-imp-act" "en-eq:3[4]" "pre-en-eq:1[4]")
3402
3403AOT_theorem "oa-facts:1": ‹O!x → □O!x›
3404proof(rule "→I")
3405  AOT_modally_strict {
3406    AOT_have ‹[λx ◇E!x]x ≡ ◇E!x›
3407      by (rule "lambda-predicates:2"[axiom_inst, THEN "→E"]) "cqt:2[lambda]"
3408  } note θ = this
3409  AOT_assume ‹O!x›
3410  AOT_hence ‹[λx ◇E!x]x›
3411    by (rule "=dfE"(2)[OF AOT_ordinary, rotated 1]) "cqt:2[lambda]"
3412  AOT_hence ‹◇E!x› using θ[THEN "≡E"(1)] by blast
3413  AOT_hence 0: ‹□◇E!x› using "qml:3"[axiom_inst, THEN "→E"] by blast
3414  AOT_have ‹□[λx ◇E!x]x›
3415    by (AOT_subst_using subst: θ) (simp add: 0)
3416  AOT_thus ‹□O!x›
3417    by (rule "=dfI"(2)[OF AOT_ordinary, rotated 1]) "cqt:2[lambda]"
3418qed
3419
3420AOT_theorem "oa-facts:2": ‹A!x → □A!x›
3421proof(rule "→I")
3422  AOT_modally_strict {
3423    AOT_have ‹[λx ¬◇E!x]x ≡ ¬◇E!x›
3424      by (rule "lambda-predicates:2"[axiom_inst, THEN "→E"]) "cqt:2[lambda]"
3425  } note θ = this
3426  AOT_assume ‹A!x›
3427  AOT_hence ‹[λx ¬◇E!x]x›
3428    by (rule "=dfE"(2)[OF AOT_abstract, rotated 1]) "cqt:2[lambda]"
3429  AOT_hence ‹¬◇E!x› using θ[THEN "≡E"(1)] by blast
3430  AOT_hence ‹□¬E!x› using "KBasic2:1"[THEN "≡E"(2)] by blast
3431  AOT_hence 0: ‹□□¬E!x› using "4"[THEN "→E"] by blast
3432  AOT_have 1: ‹□¬◇E!x›
3433    apply (AOT_subst "«¬◇E!x»" "«□¬E!x»")
3434    using "KBasic2:1"[symmetric] apply blast
3435    using 0 by blast
3436  AOT_have ‹□[λx ¬◇E!x]x›
3437    by (AOT_subst_using subst: θ) (simp add: 1)
3438  AOT_thus ‹□A!x›
3439    by (rule "=dfI"(2)[OF AOT_abstract, rotated 1]) "cqt:2[lambda]"
3440qed
3441
3442AOT_theorem "oa-facts:3": ‹◇O!x → O!x›
3443  using "oa-facts:1" "B◇" "RM◇" "Hypothetical Syllogism" by blast
3444AOT_theorem "oa-facts:4": ‹◇A!x → A!x›
3445  using "oa-facts:2" "B◇" "RM◇" "Hypothetical Syllogism" by blast
3446
3447AOT_theorem "oa-facts:5": ‹◇O!x ≡ □O!x›
3448  by (meson "Act-Sub:3" "Hypothetical Syllogism" "≡I" "nec-imp-act" "oa-facts:1" "oa-facts:3")
3449
3450AOT_theorem "oa-facts:6": ‹◇A!x ≡ □A!x›
3451  by (meson "Act-Sub:3" "Hypothetical Syllogism" "≡I" "nec-imp-act" "oa-facts:2" "oa-facts:4")
3452
3453AOT_theorem "oa-facts:7": ‹O!x ≡ 𝒜O!x›
3454  by (meson "Act-Sub:3" "Hypothetical Syllogism" "≡I" "nec-imp-act" "oa-facts:1" "oa-facts:3")
3455
3456AOT_theorem "oa-facts:8": ‹A!x ≡ 𝒜A!x›
3457  by (meson "Act-Sub:3" "Hypothetical Syllogism" "≡I" "nec-imp-act" "oa-facts:2" "oa-facts:4")
3458
3459AOT_theorem "beta-C-meta": ‹[λμ1...μn φ{μ1...μn, ν1...νn}]↓ → ([λμ1...μn φ{μ1...μn, ν1...νn}]ν1...νn ≡ φ{ν1...νn, ν1...νn})›
3460  using "lambda-predicates:2"[axiom_inst] by blast
3461
3462AOT_theorem "beta-C-cor:1": ‹(∀ν1...∀νn([λμ1...μn φ{μ1...μn, ν1...νn}]↓)) → ∀ν1...∀νn ([λμ1...μn φ{μ1...μn, ν1...νn}]ν1...νn ≡ φ{ν1...νn, ν1...νn})›
3463  apply (rule "cqt-basic:14"[where 'a='a, THEN "→E"])
3464  using "beta-C-meta" "∀I" by fast
3465
3466AOT_theorem "beta-C-cor:2": ‹[λμ1...μn φ{μ1...μn}]↓ → ∀ν1...∀νn ([λμ1...μn φ{μ1...μn}]ν1...νn ≡ φ{ν1...νn})›
3467  apply (rule "→I"; rule "∀I")
3468  using "beta-C-meta"[THEN "→E"] by fast
3469
3470(* TODO: syntax + double-check if this is really a faithful representation *)
3471theorem "beta-C-cor:3": assumes ‹⋀ν1νn. AOT_instance_of_cqt_2 (φ (AOT_term_of_var ν1νn))›
3472  shows ‹[v ⊨ ∀ν1...∀νn ([λμ1...μn φ{ν1...νn,μ1...μn}]ν1...νn ≡ φ{ν1...νn,ν1...νn})]›
3473  using "cqt:2[lambda]"[axiom_inst, OF assms] "beta-C-cor:1"[THEN "→E"] "∀I" by fast
3474
3475AOT_theorem "betaC:1:a": ‹[λμ1...μn φ{μ1...μn}]κ1...κn ⊢□ φ{κ1...κn}›
3476proof -
3477  AOT_modally_strict {
3478    AOT_assume ‹[λμ1...μn φ{μ1...μn}]κ1...κn›
3479    moreover AOT_have ‹[λμ1...μn φ{μ1...μn}]↓› and ‹κ1...κn↓›
3480      using calculation "cqt:5:a"[axiom_inst, THEN "→E"] "&E" by blast+
3481    ultimately AOT_show ‹φ{κ1...κn}›
3482      using "beta-C-cor:2"[THEN "→E", THEN "∀E"(1), THEN "≡E"(1)] by blast
3483  }
3484qed
3485
3486AOT_theorem "betaC:1:b": ‹¬φ{κ1...κn} ⊢□ ¬[λμ1...μn φ{μ1...μn}]κ1...κn›
3487  using "betaC:1:a" "raa-cor:3" by blast
3488
3489lemmas "β→C" = "betaC:1:a" "betaC:1:b"
3490
3491AOT_theorem "betaC:2:a": ‹[λμ1...μn φ{μ1...μn}]↓, κ1...κn↓, φ{κ1...κn} ⊢□ [λμ1...μn φ{μ1...μn}]κ1...κn›
3492proof -
3493  AOT_modally_strict {
3494    AOT_assume 1: ‹[λμ1...μn φ{μ1...μn}]↓› and 2: ‹κ1...κn↓› and 3: ‹φ{κ1...κn}›
3495    AOT_hence ‹[λμ1...μn φ{μ1...μn}]κ1...κn›
3496      using "beta-C-cor:2"[THEN "→E", OF 1, THEN "∀E"(1), THEN "≡E"(2)] by blast
3497  }
3498  AOT_thus ‹[λμ1...μn φ{μ1...μn}]↓, κ1...κn↓, φ{κ1...κn} ⊢□ [λμ1...μn φ{μ1...μn}]κ1...κn›
3499    by blast
3500qed
3501
3502AOT_theorem "betaC:2:b": ‹[λμ1...μn φ{μ1...μn}]↓, κ1...κn↓, ¬[λμ1...μn φ{μ1...μn}]κ1...κn ⊢□ ¬φ{κ1...κn}›
3503  using "betaC:2:a" "raa-cor:3" by blast
3504
3505lemmas "β←C" = "betaC:2:a" "betaC:2:b"
3506
3507AOT_theorem "eta-conversion-lemma1:1": ‹Π↓ → [λx1...xn [Π]x1...xn] = Π›
3508  using "lambda-predicates:3"[axiom_inst] "∀I" "∀E"(1) "→I" by fast
3509
3510AOT_theorem "eta-conversion-lemma1:2": ‹Π↓ → [λν1...νn [Π]ν1...νn] = Π›
3511  using "eta-conversion-lemma1:1". (* TODO: spurious in the embedding *)
3512
3513(* match (τ) in "λa . ?b" ⇒ ‹match (τ') in "λa . ?b" ⇒ ‹fail›› ¦ _ ⇒ ‹ *)
3514
3515text‹Note: not explicitly part of PLM.›
3516AOT_theorem id_sym: assumes ‹τ = τ'› shows ‹τ' = τ›
3517  using "rule=E"[where φ="λ τ' . «τ' = τ»", rotated 1, OF assms]
3518        "=I"(1)[OF "t=t-proper:1"[THEN "→E", OF assms]] by auto
3519declare id_sym[sym]
3520
3521text‹Note: not explicitly part of PLM.›
3522AOT_theorem id_trans: assumes ‹τ = τ'› and ‹τ' = τ''› shows ‹τ = τ''›
3523  using "rule=E" assms by blast
3524declare id_trans[trans]
3525
3526method "ηC" for Π :: ‹<'a::{AOT_Term_id_2,AOT_κs}>› = (match conclusion in "[v ⊨ τ{Π} = τ'{Π}]" for v τ τ' ⇒ ‹
3527rule "rule=E"[rotated 1, OF "eta-conversion-lemma1:2"[THEN "→E", of v "«[Π]»", symmetric]]
3528›)
3529(*
3530AOT_theorem ‹[λy [λz [P]z]y → [λu [S]u]y] = [λy [P]y → [S]y]›
3531  apply ("ηC" "«[P]»") defer
3532   apply ("ηC" "«[S]»") defer
3533  oops
3534*)
3535(* TODO: proper representation of eta_conversion_lemma2 *)
3536
3537AOT_theorem "sub-des-lam:1": ‹[λz1...zn  χ{z1...zn, ιx φ{x}}]↓ & ιx φ{x} = ιx ψ{x} → [λz1...zn χ{z1...zn, ιx φ{x}}] = [λz1...zn χ{z1...zn, ιx ψ{x}}]›
3538proof(rule "→I")
3539  AOT_assume A: ‹[λz1...zn  χ{z1...zn, ιx φ{x}}]↓ & ιx φ{x} = ιx ψ{x}›
3540  AOT_show ‹[λz1...zn χ{z1...zn, ιx φ{x}}] = [λz1...zn χ{z1...zn, ιx ψ{x}}]›
3541    using "rule=E"[where φ="λ τ . «[λz1...zn χ{z1...zn, ιx φ{x}}] = [λz1...zn χ{z1...zn, τ}]»",
3542               OF "=I"(1)[OF A[THEN "&E"(1)]], OF A[THEN "&E"(2)]]
3543    by blast
3544qed
3545
3546AOT_theorem "sub-des-lam:2": ‹ιx φ{x} = ιx ψ{x} → χ{ιx φ{x}} = χ{ιx ψ{x}}› for χ :: ‹κ ⇒ 𝗈›
3547  using "rule=E"[where φ="λ τ . «χ{ιx φ{x}} = χ{τ}»", OF "=I"(1)[OF "log-prop-prop:2"]] "→I" by blast
3548
3549AOT_theorem "prop-equiv": ‹F = G ≡ ∀x (x[F] ≡ x[G])›
3550proof(rule "≡I"; rule "→I")
3551  AOT_assume ‹F = G›
3552  AOT_thus ‹∀x (x[F] ≡ x[G])›
3553    by (rule "rule=E"[rotated]) (fact "oth-class-taut:3:a"[THEN GEN])
3554next
3555  AOT_assume ‹∀x (x[F] ≡ x[G])›
3556  AOT_hence ‹x[F] ≡ x[G]› for x using "∀E" by blast
3557  AOT_hence ‹□(x[F] ≡ x[G])› for x using "en-eq:6[1]"[THEN "≡E"(1)] by blast
3558  AOT_hence ‹∀x □(x[F] ≡ x[G])› by (rule GEN)
3559  AOT_hence ‹□∀x (x[F] ≡ x[G])› using BF[THEN "→E"] by fast
3560  AOT_thus "F = G" using "p-identity-thm2:1"[THEN "≡E"(2)] by blast
3561qed
3562
3563AOT_theorem "relations:1":
3564  assumes ‹INSTANCE_OF_CQT_2(φ)›
3565  shows ‹∃F □∀x1...∀xn ([F]x1...xn ≡ φ{x1...xn})›
3566  apply (rule "∃I"(1)[where τ="«[λx1...xn φ{x1...xn}]»"])
3567  using "cqt:2[lambda]"[OF assms, axiom_inst] "beta-C-cor:2"[THEN "→E", THEN RN] by blast+
3568
3569AOT_theorem "relations:2":
3570  assumes ‹INSTANCE_OF_CQT_2(φ)›
3571  shows ‹∃F □∀x ([F]x ≡ φ{x})›
3572  using "relations:1" assms by blast
3573
3574AOT_theorem "block-paradox:1": ‹¬[λx ∃G (x[G] & ¬[G]x)]↓›
3575proof(rule RAA(2))
3576  let ?φ="λ τ. «∃G (τ[G] & ¬[G]τ)»"
3577  AOT_assume A: ‹[λx «?φ x»]↓›
3578  AOT_have ‹∃x (A!x & ∀F (x[F] ≡ F = [λx «?φ x»]))›
3579    using "A-objects"[axiom_inst] by fast
3580  then AOT_obtain a where ξ: ‹A!a & ∀F (a[F] ≡ F = [λx «?φ x»])›
3581    using "∃E"[rotated] by blast
3582  AOT_show ‹¬[λx ∃G (x[G] & ¬[G]x)]↓›
3583  proof (rule "∨E"(1)[OF "exc-mid"]; rule "→I")
3584    AOT_assume B: ‹[λx «?φ x»]a›
3585    AOT_hence ‹∃G (a[G] & ¬[G]a)› using "β→C" A by blast
3586    then AOT_obtain P where ‹a[P] & ¬[P]a› using "∃E"[rotated] by blast
3587    moreover AOT_have ‹P = [λx «?φ x»]›
3588      using ξ[THEN "&E"(2), THEN "∀E"(2), THEN "≡E"(1)] calculation[THEN "&E"(1)] by blast
3589    ultimately AOT_have ‹¬[λx «?φ x»]a›
3590      using "rule=E" "&E"(2) by fast
3591    AOT_thus ‹¬[λx ∃G (x[G] & ¬[G]x)]↓› using B RAA by blast
3592  next
3593    AOT_assume B: ‹¬[λx «?φ x»]a›
3594    AOT_hence ‹¬∃G (a[G] & ¬[G]a)› using "β←C" "cqt:2[const_var]"[of a, axiom_inst] A by blast
3595    AOT_hence C: ‹∀G ¬(a[G] & ¬[G]a)› using "cqt-further:4"[THEN "→E"] by blast
3596    AOT_have ‹∀G (a[G] → [G]a)›
3597      by (AOT_subst "λ Π . «a[Π] → [Π]a»" "λ Π . «¬(a[Π] & ¬[Π]a)»")
3598         (auto simp: "oth-class-taut:1:a" C)
3599    AOT_hence ‹a[λx «?φ x»] → [λx «?φ x»]a› using "∀E" A by blast
3600    moreover AOT_have ‹a[λx «?φ x»]› using ξ[THEN "&E"(2), THEN "∀E"(1), OF A, THEN "≡E"(2)]
3601      using "=I"(1)[OF A] by blast
3602    ultimately AOT_show ‹¬[λx ∃G (x[G] & ¬[G]x)]↓› using B "→E" RAA by blast
3603  qed
3604qed(simp)
3605
3606AOT_theorem "block-paradox:2": ‹¬∃F ∀x([F]x ≡ ∃G(x[G] & ¬[G]x))›
3607proof(rule RAA(2))
3608  AOT_assume ‹∃F ∀x ([F]x ≡ ∃G (x[G] & ¬[G]x))›
3609  then AOT_obtain F where F_prop: ‹∀x ([F]x ≡ ∃G (x[G] & ¬[G]x))› using "∃E"[rotated] by blast
3610  AOT_have ‹∃x (A!x & ∀G (x[G] ≡ G = F))›
3611    using "A-objects"[axiom_inst] by fast
3612  then AOT_obtain a where ξ: ‹A!a & ∀G (a[G] ≡ G = F)›
3613    using "∃E"[rotated] by blast
3614  AOT_show ‹¬∃F ∀x([F]x ≡ ∃G(x[G] & ¬[G]x))›
3615  proof (rule "∨E"(1)[OF "exc-mid"]; rule "→I")
3616    AOT_assume B: ‹[F]a›
3617    AOT_hence ‹∃G (a[G] & ¬[G]a)› using F_prop[THEN "∀E"(2), THEN "≡E"(1)] by blast
3618    then AOT_obtain P where ‹a[P] & ¬[P]a› using "∃E"[rotated] by blast
3619    moreover AOT_have ‹P = F›
3620      using ξ[THEN "&E"(2), THEN "∀E"(2), THEN "≡E"(1)] calculation[THEN "&E"(1)] by blast
3621    ultimately AOT_have ‹¬[F]a›
3622      using "rule=E" "&E"(2) by fast
3623    AOT_thus ‹¬∃F ∀x([F]x ≡ ∃G(x[G] & ¬[G]x))› using B RAA by blast
3624  next
3625    AOT_assume B: ‹¬[F]a›
3626    AOT_hence ‹¬∃G (a[G] & ¬[G]a)›
3627      using "oth-class-taut:4:b"[THEN "≡E"(1), OF F_prop[THEN "∀E"(2)[of _ _ a]], THEN "≡E"(1)] by simp
3628    AOT_hence C: ‹∀G ¬(a[G] & ¬[G]a)› using "cqt-further:4"[THEN "→E"] by blast
3629    AOT_have ‹∀G (a[G] → [G]a)›
3630      by (AOT_subst "λ Π . «a[Π] → [Π]a»" "λ Π . «¬(a[Π] & ¬[Π]a)»")
3631         (auto simp: "oth-class-taut:1:a" C)
3632    AOT_hence ‹a[F] → [F]a› using "∀E" by blast
3633    moreover AOT_have ‹a[F]› using ξ[THEN "&E"(2), THEN "∀E"(2), of F, THEN "≡E"(2)]
3634      using "=I"(2) by blast
3635    ultimately AOT_show ‹¬∃F ∀x([F]x ≡ ∃G(x[G] & ¬[G]x))› using B "→E" RAA by blast
3636  qed
3637qed(simp)
3638
3639AOT_theorem "block-paradox:3": ‹¬∀y [λz z = y]↓›
3640proof(rule RAA(2))
3641  AOT_assume θ: ‹∀y [λz z = y]↓›
3642  AOT_have ‹∃x (A!x & ∀F (x[F] ≡ ∃y(F = [λz z = y] & ¬y[F])))›
3643    using "A-objects"[axiom_inst] by force
3644  then AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ ∃y (F = [λz z = y] & ¬y[F]))›
3645    using "∃E"[rotated] by blast
3646  AOT_have ζ: ‹a[λz z = a] ≡ ∃y ([λz z = a] = [λz z = y] & ¬y[λz z = a])›
3647    using θ[THEN "∀E"(2)] a_prop[THEN "&E"(2), THEN "∀E"(1)] by blast
3648  AOT_show ‹¬∀y [λz z = y]↓›
3649  proof (rule "∨E"(1)[OF "exc-mid"]; rule "→I")
3650    AOT_assume A: ‹a[λz z = a]›
3651    AOT_hence ‹∃y ([λz z = a] = [λz z = y] & ¬y[λz z = a])›
3652      using ζ[THEN "≡E"(1)] by blast
3653    then AOT_obtain b where b_prop: ‹[λz z = a] = [λz z = b] & ¬b[λz z = a]›
3654      using "∃E"[rotated] by blast
3655    moreover AOT_have ‹a = a› by (rule "=I")
3656    moreover AOT_have ‹[λz z = a]↓› using θ "∀E" by blast
3657    moreover AOT_have ‹a↓› using "cqt:2[const_var]"[axiom_inst] .
3658    ultimately AOT_have ‹[λz z = a]a› using "β←C" by blast
3659    AOT_hence ‹[λz z = b]a› using "rule=E" b_prop[THEN "&E"(1)] by fast
3660    AOT_hence ‹a = b› using "β→C" by blast
3661    AOT_hence ‹b[λz z = a]› using A "rule=E" by fast
3662    AOT_thus ‹¬∀y [λz z = y]↓› using b_prop[THEN "&E"(2)] RAA by blast
3663  next
3664    AOT_assume A: ‹¬a[λz z = a]›
3665    AOT_hence ‹¬∃y ([λz z = a] = [λz z = y] & ¬y[λz z = a])›
3666      using ζ "oth-class-taut:4:b"[THEN "≡E"(1), THEN "≡E"(1)] by blast
3667    AOT_hence ‹∀y ¬([λz z = a] = [λz z = y] & ¬y[λz z = a])›
3668      using "cqt-further:4"[THEN "→E"] by blast
3669    AOT_hence ‹¬([λz z = a] = [λz z = a] & ¬a[λz z = a])›
3670      using "∀E" by blast
3671    AOT_hence ‹[λz z = a] = [λz z = a] → a[λz z = a]›
3672      by (metis "&I" "deduction-theorem" "raa-cor:4")
3673    AOT_hence ‹a[λz z = a]› using "=I"(1) θ[THEN "∀E"(2)] "→E" by blast
3674    AOT_thus ‹¬∀y [λz z = y]↓› using A RAA by blast
3675  qed
3676qed(simp)
3677
3678AOT_theorem "block-paradox:4": ‹¬∀y ∃F ∀x([F]x ≡ x = y)›
3679proof(rule RAA(2))
3680  AOT_assume θ: ‹∀y ∃F ∀x([F]x ≡ x = y)›
3681  AOT_have ‹∃x (A!x & ∀F (x[F] ≡ ∃z (∀y([F]y ≡ y = z) & ¬z[F])))›
3682    using "A-objects"[axiom_inst] by force
3683  then AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ ∃z (∀y([F]y ≡ y = z) & ¬z[F]))›
3684    using "∃E"[rotated] by blast
3685  AOT_obtain F where F_prop: ‹∀x ([F]x ≡ x = a)› using θ[THEN "∀E"(2)] "∃E"[rotated] by blast
3686  AOT_have ζ: ‹a[F] ≡ ∃z (∀y ([F]y ≡ y = z) & ¬z[F])›
3687    using a_prop[THEN "&E"(2), THEN "∀E"(2)] by blast
3688  AOT_show ‹¬∀y ∃F ∀x([F]x ≡ x = y)›
3689  proof (rule "∨E"(1)[OF "exc-mid"]; rule "→I")
3690    AOT_assume A: ‹a[F]›
3691    AOT_hence ‹∃z (∀y ([F]y ≡ y = z) & ¬z[F])›
3692      using ζ[THEN "≡E"(1)] by blast
3693    then AOT_obtain b where b_prop: ‹∀y ([F]y ≡ y = b) & ¬b[F]›
3694      using "∃E"[rotated] by blast
3695    moreover AOT_have ‹[F]a› using F_prop[THEN "∀E"(2), THEN "≡E"(2)] "=I"(2) by blast
3696    ultimately AOT_have ‹a = b› using "∀E"(2) "≡E"(1) "&E" by fast
3697    AOT_hence ‹a = b› using "β→C" by blast
3698    AOT_hence ‹b[F]› using A "rule=E" by fast
3699    AOT_thus ‹¬∀y ∃F ∀x([F]x ≡ x = y)› using b_prop[THEN "&E"(2)] RAA by blast
3700  next
3701    AOT_assume A: ‹¬a[F]›
3702    AOT_hence ‹¬∃z (∀y ([F]y ≡ y = z) & ¬z[F])›
3703      using ζ "oth-class-taut:4:b"[THEN "≡E"(1), THEN "≡E"(1)] by blast
3704    AOT_hence ‹∀z ¬(∀y ([F]y ≡ y = z) & ¬z[F])›
3705      using "cqt-further:4"[THEN "→E"] by blast
3706    AOT_hence ‹¬(∀y ([F]y ≡ y = a) & ¬a[F])›
3707      using "∀E" by blast
3708    AOT_hence ‹∀y ([F]y ≡ y = a) → a[F]›
3709      by (metis "&I" "deduction-theorem" "raa-cor:4")
3710    AOT_hence ‹a[F]› using F_prop "→E" by blast
3711    AOT_thus ‹¬∀y ∃F ∀x([F]x ≡ x = y)› using A RAA by blast
3712  qed
3713qed(simp)
3714
3715AOT_theorem "block-paradox:5": ‹¬∃F∀x∀y([F]xy ≡ y = x)›
3716proof(rule "raa-cor:2")
3717  AOT_assume ‹∃F∀x∀y([F]xy ≡ y = x)›
3718  then AOT_obtain F where F_prop: ‹∀x∀y([F]xy ≡ y = x)› using "∃E"[rotated] by blast
3719  {
3720    fix x
3721    AOT_have 1: ‹∀y([F]xy ≡ y = x)› using F_prop "∀E" by blast
3722    AOT_have 2: ‹[λz [F]xz]↓› by "cqt:2[lambda]"
3723    moreover AOT_have ‹∀y([λz [F]xz]y ≡ y = x)›
3724    proof(rule "∀I")
3725      fix y
3726      AOT_have ‹[λz [F]xz]y ≡ [F]xy›
3727        using "beta-C-meta"[THEN "→E"] 2 by fast
3728      also AOT_have ‹... ≡ y = x› using 1 "∀E"
3729        by fast
3730      finally AOT_show ‹[λz [F]xz]y ≡ y = x›.
3731    qed
3732    ultimately AOT_have ‹∃F∀y([F]y ≡ y = x)›
3733      using "∃I" by fast
3734  }
3735  AOT_hence ‹∀x∃F∀y([F]y ≡ y = x)›
3736    by (rule GEN)
3737  AOT_thus ‹∀x∃F∀y([F]y ≡ y = x) & ¬∀x∃F∀y([F]y ≡ y = x)›
3738    using "&I" "block-paradox:4" by blast
3739qed
3740
3741AOT_act_theorem "block-paradox2:1": ‹∀x [G]x → ¬[λx [G]ιy (y = x & ∃H (x[H] & ¬[H]x))]↓›
3742proof(rule "→I"; rule "raa-cor:2")
3743  AOT_assume antecedant: ‹∀x [G]x›
3744  AOT_have Lemma: ‹∀x ([G]ιy(y = x & ∃H (x[H] & ¬[H]x)) ≡ ∃H (x[H] & ¬[H]x))›
3745  proof(rule GEN)
3746    fix x
3747    AOT_have A: ‹[G]ιy (y = x & ∃H (x[H] & ¬[H]x)) ≡ ∃!y (y = x & ∃H (x[H] & ¬[H]x))›
3748    proof(rule "≡I"; rule "→I")
3749      AOT_assume ‹[G]ιy (y = x & ∃H (x[H] & ¬[H]x))›
3750      AOT_hence ‹ιy (y = x & ∃H (x[H] & ¬[H]x))↓›
3751        using "cqt:5:a"[axiom_inst, THEN "→E", THEN "&E"(2)] by blast
3752      AOT_thus ‹∃!y (y = x & ∃H (x[H] & ¬[H]x))›
3753        using "1-exists:1"[THEN "≡E"(1)] by blast
3754    next
3755      AOT_assume A: ‹∃!y (y = x & ∃H (x[H] & ¬[H]x))›
3756      AOT_obtain a where a_1: ‹a = x & ∃H (x[H] & ¬[H]x)› and a_2: ‹∀z (z = x & ∃H (x[H] & ¬[H]x) → z = a)›
3757        using "uniqueness:1"[THEN "≡dfE", OF A] "&E" "∃E"[rotated] by blast
3758      AOT_have a_3: ‹[G]a›
3759        using antecedant "∀E" by blast
3760      AOT_show ‹[G]ιy (y = x & ∃H (x[H] & ¬[H]x))›
3761        apply (rule "russell-axiom[exe,1].russell-axiom"[THEN "≡E"(2)])
3762        apply (rule "∃I"(2))
3763        using a_1 a_2 a_3 "&I" by blast
3764    qed
3765    also AOT_have B: ‹... ≡ ∃H (x[H] & ¬[H]x)›
3766    proof (rule "≡I"; rule "→I")
3767      AOT_assume A: ‹∃!y (y = x & ∃H (x[H] & ¬[H]x))›
3768      AOT_obtain a where ‹a = x & ∃H (x[H] & ¬[H]x)›
3769        using "uniqueness:1"[THEN "≡dfE", OF A] "&E" "∃E"[rotated] by blast
3770      AOT_thus ‹∃H (x[H] & ¬[H]x)› using "&E" by blast
3771    next
3772      AOT_assume ‹∃H (x[H] & ¬[H]x)›
3773      AOT_hence ‹x = x & ∃H (x[H] & ¬[H]x)›
3774        using "id-eq:1" "&I" by blast
3775      moreover AOT_have ‹∀z (z = x & ∃H (x[H] & ¬[H]x) → z = x)›
3776        by (simp add: "Conjunction Simplification"(1) "universal-cor")
3777      ultimately AOT_show ‹∃!y (y = x & ∃H (x[H] & ¬[H]x))›
3778        using "uniqueness:1"[THEN "≡dfI"] "&I" "∃I"(2) by fast
3779    qed
3780    finally AOT_show ‹([G]ιy(y = x & ∃H (x[H] & ¬[H]x)) ≡ ∃H (x[H] & ¬[H]x))› .
3781  qed
3782
3783  AOT_assume A: ‹[λx [G]ιy (y = x & ∃H (x[H] & ¬[H]x))]↓›
3784  AOT_have θ: ‹∀x ([λx [G]ιy (y = x & ∃H (x[H] & ¬[H]x))]x ≡ [G]ιy(y = x & ∃H (x[H] & ¬[H]x)))›
3785    using "beta-C-meta"[THEN "→E", OF A] "∀I" by fast
3786  AOT_have ‹∀x ([λx [G]ιy (y = x & ∃H (x[H] & ¬[H]x))]x ≡ ∃H (x[H] & ¬[H]x))›
3787    using θ Lemma "cqt-basic:10"[THEN "→E"] "&I" by fast
3788  AOT_hence ‹∃F ∀x ([F]x ≡ ∃H (x[H] & ¬[H]x))›
3789    using "∃I"(1) A by fast
3790  AOT_thus ‹(∃F ∀x ([F]x ≡ ∃H (x[H] & ¬[H]x))) & (¬∃F ∀x ([F]x ≡ ∃H (x[H] & ¬[H]x)))›
3791    using "block-paradox:2" "&I" by blast
3792qed
3793
3794AOT_act_theorem "block-paradox2:2": ‹∃G ¬[λx [G]ιy (y = x & ∃H (x[H] & ¬[H]x))]↓›
3795proof(rule "∃I"(1))
3796  AOT_have 0: ‹[λx ∀p (p →p)]↓›
3797    by "cqt:2[lambda]"
3798  moreover AOT_have ‹∀x [λx ∀p (p →p)]x›
3799    apply (rule GEN)
3800    apply (rule "beta-C-cor:2"[THEN "→E", OF 0, THEN "∀E"(2), THEN "≡E"(2)])
3801    using "if-p-then-p" GEN by fast
3802  moreover AOT_have ‹∀G (∀x [G]x → ¬[λx [G]ιy (y = x & ∃H (x[H] & ¬[H]x))]↓)›
3803      using "block-paradox2:1" "∀I" by fast
3804  ultimately AOT_show ‹¬[λx [λx ∀p (p →p)]ιy (y = x & ∃H (x[H] & ¬[H]x))]↓›
3805    using "∀E"(1) "→E" by blast
3806qed("cqt:2[lambda]")
3807
3808AOT_theorem propositions: ‹∃p □(p ≡ φ)›
3809proof(rule "∃I"(1))
3810  AOT_show ‹□(φ ≡ φ)›
3811    by (simp add: RN "oth-class-taut:3:a")
3812next
3813  AOT_show ‹φ↓›
3814    by (simp add: "log-prop-prop:2")
3815qed
3816
3817AOT_theorem "pos-not-equiv-ne:1": ‹(◇¬∀x1...∀xn ([F]x1...xn ≡ [G]x1...xn)) → F ≠ G›
3818proof (rule "→I")
3819  AOT_assume ‹◇¬∀x1...∀xn ([F]x1...xn ≡ [G]x1...xn)›
3820  AOT_hence ‹¬□∀x1...∀xn ([F]x1...xn ≡ [G]x1...xn)›
3821    using "KBasic:11"[THEN "≡E"(2)] by blast
3822  AOT_hence ‹¬(F = G)›
3823    using "id-rel-nec-equiv:1" "modus-tollens:1" by blast
3824  AOT_thus ‹F ≠ G›
3825    using "=-infix"[THEN "≡dfI"] by blast
3826qed
3827
3828AOT_theorem "pos-not-equiv-ne:2": ‹(◇¬(φ{F} ≡ φ{G})) → F ≠ G›
3829proof (rule "→I")
3830  AOT_modally_strict {
3831    AOT_have ‹¬(φ{F} ≡ φ{G}) → ¬(F = G)›
3832    proof (rule "→I"; rule "raa-cor:2")
3833      AOT_assume 1: ‹F = G›
3834      AOT_hence ‹φ{F} → φ{G}› using "l-identity"[axiom_inst, THEN "→E"] by blast
3835      moreover {
3836        AOT_have ‹G = F› using 1 id_sym by blast
3837        AOT_hence ‹φ{G} → φ{F}› using "l-identity"[axiom_inst, THEN "→E"] by blast
3838      }
3839      ultimately AOT_have ‹φ{F} ≡ φ{G}› using "≡I" by blast
3840      moreover AOT_assume ‹¬(φ{F} ≡ φ{G})›
3841      ultimately AOT_show ‹(φ{F} ≡ φ{G}) & ¬(φ{F} ≡ φ{G})›
3842        using "&I" by blast
3843    qed
3844  }
3845  AOT_hence ‹◇¬(φ{F} ≡ φ{G}) → ◇¬(F = G)›
3846    using "RM:2[prem]" by blast
3847  moreover AOT_assume ‹◇¬(φ{F} ≡ φ{G})›
3848  ultimately AOT_have 0: ‹◇¬(F = G)› using "→E" by blast
3849  AOT_have ‹◇(F ≠ G)›
3850    by (AOT_subst "«F ≠ G»" "«¬(F = G)»")
3851       (auto simp: "=-infix" "≡Df" 0)
3852  AOT_thus ‹F ≠ G›
3853    using "id-nec2:3"[THEN "→E"] by blast
3854qed
3855
3856AOT_theorem "pos-not-equiv-ne:2[zero]": ‹(◇¬(φ{p} ≡ φ{q})) → p ≠ q›
3857proof (rule "→I")
3858  AOT_modally_strict {
3859    AOT_have ‹¬(φ{p} ≡ φ{q}) → ¬(p = q)›
3860    proof (rule "→I"; rule "raa-cor:2")
3861      AOT_assume 1: ‹p = q›
3862      AOT_hence ‹φ{p} → φ{q}› using "l-identity"[axiom_inst, THEN "→E"] by blast
3863      moreover {
3864        AOT_have ‹q = p› using 1 id_sym by blast
3865        AOT_hence ‹φ{q} → φ{p}› using "l-identity"[axiom_inst, THEN "→E"] by blast
3866      }
3867      ultimately AOT_have ‹φ{p} ≡ φ{q}› using "≡I" by blast
3868      moreover AOT_assume ‹¬(φ{p} ≡ φ{q})›
3869      ultimately AOT_show ‹(φ{p} ≡ φ{q}) & ¬(φ{p} ≡ φ{q})›
3870        using "&I" by blast
3871    qed
3872  }
3873  AOT_hence ‹◇¬(φ{p} ≡ φ{q}) → ◇¬(p = q)›
3874    using "RM:2[prem]" by blast
3875  moreover AOT_assume ‹◇¬(φ{p} ≡ φ{q})›
3876  ultimately AOT_have 0: ‹◇¬(p = q)› using "→E" by blast
3877  AOT_have ‹◇(p ≠ q)›
3878    by (AOT_subst "«p ≠ q»" "«¬(p = q)»")
3879       (auto simp: 0 "=-infix" "≡Df")
3880  AOT_thus ‹p ≠ q›
3881    using "id-nec2:3"[THEN "→E"] by blast
3882qed
3883
3884AOT_theorem "pos-not-equiv-ne:3": ‹(¬∀x1...∀xn ([F]x1...xn ≡ [G]x1...xn)) → F ≠ G›
3885  using "→I" "pos-not-equiv-ne:1"[THEN "→E"] "T◇"[THEN "→E"] by blast
3886
3887AOT_theorem "pos-not-equiv-ne:4": ‹(¬(φ{F} ≡ φ{G})) → F ≠ G›
3888  using "→I" "pos-not-equiv-ne:2"[THEN "→E"] "T◇"[THEN "→E"] by blast
3889
3890AOT_theorem "pos-not-equiv-ne:4[zero]": ‹(¬(φ{p} ≡ φ{q})) → p ≠ q›
3891  using "→I" "pos-not-equiv-ne:2[zero]"[THEN "→E"] "T◇"[THEN "→E"] by blast
3892
3893AOT_define relation_negation :: "Π ⇒ Π" ("_-")
3894  "df-relation-negation": "[F]- =df [λx1...xn ¬[F]x1...xn]"
3895
3896nonterminal φneg
3897syntax "" :: "φneg ⇒ τ" ("_")
3898syntax "" :: "φneg ⇒ φ" ("'(_')")
3899
3900AOT_define relation_negation_0 :: ‹φ ⇒ φneg› ("'(_')-")
3901  "df-relation-negation[zero]": "(p)- =df [λ ¬p]"
3902
3903AOT_theorem "rel-neg-T:1": ‹[λx1...xn ¬[Π]x1...xn]↓›
3904  by "cqt:2[lambda]"
3905
3906AOT_theorem "rel-neg-T:1[zero]": ‹[λ ¬φ]↓›
3907  using "cqt:2[lambda0]"[axiom_inst] by blast
3908
3909AOT_theorem "rel-neg-T:2": ‹[Π]- = [λx1...xn ¬[Π]x1...xn]›
3910  using "=I"(1)[OF "rel-neg-T:1"]
3911  by (rule "=dfI"(1)[OF "df-relation-negation", OF "rel-neg-T:1"])
3912
3913AOT_theorem "rel-neg-T:2[zero]": ‹(φ)- = [λ ¬φ]›
3914  using "=I"(1)[OF "rel-neg-T:1[zero]"]
3915  by (rule "=dfI"(1)[OF "df-relation-negation[zero]", OF "rel-neg-T:1[zero]"])
3916
3917AOT_theorem "rel-neg-T:3": ‹[Π]-↓›
3918  using "=dfI"(1)[OF "df-relation-negation", OF "rel-neg-T:1"] "rel-neg-T:1" by blast
3919
3920AOT_theorem "rel-neg-T:3[zero]": ‹(φ)-↓›
3921  using "log-prop-prop:2" by blast
3922(*  using "=dfI"(1)[OF "df-relation-negation[zero]", OF "rel-neg-T:1[zero]"] "rel-neg-T:1[zero]" by blast *)
3923
3924(* Note: PLM states the zero place case twice *)
3925AOT_theorem "thm-relation-negation:1": ‹[F]-x1...xn ≡ ¬[F]x1...xn›
3926proof -
3927  AOT_have ‹[F]-x1...xn ≡ [λx1...xn ¬[F]x1...xn]x1...xn›
3928    using "rule=E"[rotated, OF "rel-neg-T:2"] "rule=E"[rotated, OF "rel-neg-T:2"[THEN id_sym]]
3929    "→I" "≡I" by fast
3930  also AOT_have ‹... ≡ ¬[F]x1...xn›
3931    using "beta-C-meta"[THEN "→E", OF "rel-neg-T:1"] by fast
3932  finally show ?thesis.
3933qed
3934
3935AOT_theorem "thm-relation-negation:2": ‹¬[F]-x1...xn ≡ [F]x1...xn›
3936  apply (AOT_subst "«[F]x1...xn»" "«¬¬[F]x1...xn»")
3937   apply (simp add: "oth-class-taut:3:b")
3938  apply (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
3939  using "thm-relation-negation:1".
3940
3941AOT_theorem "thm-relation-negation:3": ‹((p)-) ≡ ¬p›
3942proof -
3943  AOT_have ‹(p)- = [λ ¬p]› using "rel-neg-T:2[zero]" by blast
3944  AOT_hence ‹((p)-) ≡ [λ ¬p]›
3945    using "df-relation-negation[zero]" "log-prop-prop:2" "oth-class-taut:3:a" "rule-id-df:2:a" by blast
3946  also AOT_have ‹[λ ¬p] ≡ ¬p›
3947    by (simp add: "propositions-lemma:2")
3948  finally show ?thesis.
3949qed
3950
3951AOT_theorem "thm-relation-negation:4": ‹(¬((p)-)) ≡ p›
3952  using "thm-relation-negation:3"[THEN "≡E"(1)]
3953        "thm-relation-negation:3"[THEN "≡E"(2)]
3954        "≡I" "→I" RAA by metis
3955
3956AOT_theorem "thm-relation-negation:5": ‹[F] ≠ [F]-›
3957proof -
3958  AOT_have ‹¬([F] = [F]-)›
3959  proof (rule RAA(2))
3960    AOT_show ‹[F]x1...xn → [F]x1...xn› for x1xn
3961      using "if-p-then-p".
3962  next
3963    AOT_assume ‹[F] = [F]-›
3964    AOT_hence ‹[F]- = [F]› using id_sym by blast
3965    AOT_hence ‹[F]x1...xn ≡ ¬[F]x1...xn› for x1xn
3966      using "rule=E" "thm-relation-negation:1" by fast
3967    AOT_thus ‹¬([F]x1...xn → [F]x1...xn)› for x1xn
3968      using "≡E" RAA by metis
3969  qed
3970  thus ?thesis
3971    using "≡dfI" "=-infix" by blast
3972qed
3973
3974AOT_theorem "thm-relation-negation:6": ‹p ≠ (p)-›
3975proof -
3976  AOT_have ‹¬(p = (p)-)›
3977  proof (rule RAA(2))
3978    AOT_show ‹p → p›
3979      using "if-p-then-p".
3980  next
3981    AOT_assume ‹p = (p)-›
3982    AOT_hence ‹(p)- = p› using id_sym by blast
3983    AOT_hence ‹p ≡ ¬p›
3984      using "rule=E" "thm-relation-negation:3" by fast
3985    AOT_thus ‹¬(p → p)›
3986      using "≡E" RAA by metis
3987  qed
3988  thus ?thesis
3989    using "≡dfI" "=-infix" by blast
3990qed
3991
3992AOT_theorem "thm-relation-negation:7": ‹(p)- = (¬p)›
3993  apply (rule "df-relation-negation[zero]"[THEN "=dfE"(1)])
3994  using "cqt:2[lambda0]"[axiom_inst] "rel-neg-T:2[zero]" "propositions-lemma:1" id_trans by blast+
3995
3996AOT_theorem "thm-relation-negation:8": ‹p = q → (¬p) = (¬q)›
3997proof(rule "→I")
3998  AOT_assume ‹p = q›
3999  moreover AOT_have ‹(¬p)↓› using "log-prop-prop:2".
4000  moreover AOT_have ‹(¬p) = (¬p)› using calculation(2) "=I" by blast
4001  ultimately AOT_show ‹(¬p) = (¬q)›
4002    using "rule=E" by fast
4003qed
4004
4005AOT_theorem "thm-relation-negation:9": ‹p = q → (p)- = (q)-›
4006proof(rule "→I")
4007  AOT_assume ‹p = q›
4008  AOT_hence ‹(¬p) = (¬q)› using "thm-relation-negation:8" "→E" by blast
4009  AOT_thus ‹(p)- = (q)-›
4010    using "thm-relation-negation:7" id_sym id_trans by metis
4011qed
4012
4013AOT_define Necessary :: ‹Π ⇒ φ› ("Necessary'(_')")
4014  "contingent-properties:1": ‹Necessary([F]) ≡df □∀x1...∀xn [F]x1...xn›
4015
4016AOT_define Necessary0 :: ‹φ ⇒ φ› ("Necessary0'(_')")
4017  "contingent-properties:1[zero]": ‹Necessary0(p) ≡df □p›
4018
4019AOT_define Impossible :: ‹Π ⇒ φ› ("Impossible'(_')")
4020  "contingent-properties:2": ‹Impossible([F]) ≡df F↓ & □∀x1...∀xn ¬[F]x1...xn›
4021
4022AOT_define Impossible0 :: ‹φ ⇒ φ› ("Impossible0'(_')")
4023  "contingent-properties:2[zero]": ‹Impossible0(p) ≡df □¬p›
4024
4025AOT_define NonContingent :: ‹Π ⇒ φ› ("NonContingent'(_')")
4026  "contingent-properties:3": ‹NonContingent([F]) ≡df Necessary([F]) ∨ Impossible([F])›
4027
4028AOT_define NonContingent0 :: ‹φ ⇒ φ› ("NonContingent0'(_')")
4029  "contingent-properties:3[zero]": ‹NonContingent0(p) ≡df Necessary0(p) ∨ Impossible0(p)›
4030
4031AOT_define Contingent :: ‹Π ⇒ φ› ("Contingent'(_')")
4032  "contingent-properties:4": ‹Contingent([F]) ≡df F↓ & ¬(Necessary([F]) ∨ Impossible([F]))›
4033
4034AOT_define Contingent0 :: ‹φ ⇒ φ› ("Contingent0'(_')")
4035  "contingent-properties:4[zero]": ‹Contingent0(p) ≡df ¬(Necessary0(p) ∨ Impossible0(p))›
4036
4037
4038AOT_theorem "thm-cont-prop:1": ‹NonContingent([F]) ≡ NonContingent([F]-)›
4039proof (rule "≡I"; rule "→I")
4040  AOT_assume ‹NonContingent([F])›
4041  AOT_hence ‹Necessary([F]) ∨ Impossible([F])›
4042    using "≡dfE"[OF "contingent-properties:3"] by blast
4043  moreover {
4044    AOT_assume ‹Necessary([F])›
4045    AOT_hence ‹□(∀x1...∀xn [F]x1...xn)›
4046      using "≡dfE"[OF "contingent-properties:1"] by blast
4047    moreover AOT_modally_strict {
4048      AOT_assume ‹∀x1...∀xn [F]x1...xn›
4049      AOT_hence ‹[F]x1...xn› for x1xn using "∀E" by blast
4050      AOT_hence ‹¬[F]-x1...xn› for x1xn
4051        by (meson "≡E"(6) "oth-class-taut:3:a" "thm-relation-negation:2" "≡E"(1))
4052      AOT_hence ‹∀x1...∀xn ¬[F]-x1...xn› using "∀I" by fast
4053    }
4054    ultimately AOT_have ‹□(∀x1...∀xn ¬[F]-x1...xn)›
4055      using "RN[prem]"[where Γ="{«∀x1...∀xn [F]x1...xn»}", simplified] by blast
4056    AOT_hence ‹Impossible([F]-)›
4057      using "≡Df"[OF "contingent-properties:2", THEN "≡S"(1), OF "rel-neg-T:3", THEN "≡E"(2)]
4058      by blast
4059  }
4060  moreover {
4061    AOT_assume ‹Impossible([F])›
4062    AOT_hence ‹□(∀x1...∀xn ¬[F]x1...xn)›
4063      using "≡Df"[OF "contingent-properties:2", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst], THEN "≡E"(1)]
4064      by blast
4065    moreover AOT_modally_strict {
4066      AOT_assume ‹∀x1...∀xn ¬[F]x1...xn›
4067      AOT_hence ‹¬[F]x1...xn› for x1xn using "∀E" by blast
4068      AOT_hence ‹[F]-x1...xn› for x1xn
4069        by (meson "≡E"(6) "oth-class-taut:3:a" "thm-relation-negation:1" "≡E"(1))
4070      AOT_hence ‹∀x1...∀xn [F]-x1...xn› using "∀I" by fast
4071    }
4072    ultimately AOT_have ‹□(∀x1...∀xn [F]-x1...xn)›
4073      using "RN[prem]"[where Γ="{«∀x1...∀xn ¬[F]x1...xn»}"] by blast
4074    AOT_hence ‹Necessary([F]-)›
4075      using "≡dfI"[OF "contingent-properties:1"] by blast
4076  }
4077  ultimately AOT_have ‹Necessary([F]-) ∨ Impossible([F]-)›
4078    using "∨E"(1) "∨I" "→I" by metis
4079  AOT_thus ‹NonContingent([F]-)›
4080    using "≡dfI"[OF "contingent-properties:3"] by blast
4081next
4082  AOT_assume ‹NonContingent([F]-)›
4083  AOT_hence ‹Necessary([F]-) ∨ Impossible([F]-)›
4084    using "≡dfE"[OF "contingent-properties:3"] by blast
4085  moreover {
4086    AOT_assume ‹Necessary([F]-)›
4087    AOT_hence ‹□(∀x1...∀xn [F]-x1...xn)›
4088      using "≡dfE"[OF "contingent-properties:1"] by blast
4089    moreover AOT_modally_strict {
4090      AOT_assume ‹∀x1...∀xn [F]-x1...xn›
4091      AOT_hence ‹[F]-x1...xn› for x1xn using "∀E" by blast
4092      AOT_hence ‹¬[F]x1...xn› for x1xn
4093        by (meson "≡E"(6) "oth-class-taut:3:a" "thm-relation-negation:1" "≡E"(2))
4094      AOT_hence ‹∀x1...∀xn ¬[F]x1...xn› using "∀I" by fast
4095    }
4096    ultimately AOT_have ‹□∀x1...∀xn ¬[F]x1...xn›
4097      using "RN[prem]"[where Γ="{«∀x1...∀xn [F]-x1...xn»}"] by blast
4098    AOT_hence ‹Impossible([F])›
4099      using "≡Df"[OF "contingent-properties:2", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst], THEN "≡E"(2)]
4100      by blast
4101  }
4102  moreover {
4103    AOT_assume ‹Impossible([F]-)›
4104    AOT_hence ‹□(∀x1...∀xn ¬[F]-x1...xn)›
4105      using "≡Df"[OF "contingent-properties:2", THEN "≡S"(1), OF "rel-neg-T:3", THEN "≡E"(1)]
4106      by blast
4107    moreover AOT_modally_strict {
4108      AOT_assume ‹∀x1...∀xn ¬[F]-x1...xn›
4109      AOT_hence ‹¬[F]-x1...xn› for x1xn using "∀E" by blast
4110      AOT_hence ‹[F]x1...xn› for x1xn 
4111        using "thm-relation-negation:1"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1)]
4112              "useful-tautologies:1"[THEN "→E"] by blast
4113      AOT_hence ‹∀x1...∀xn [F]x1...xn› using "∀I" by fast
4114    }
4115    ultimately AOT_have ‹□(∀x1...∀xn [F]x1...xn)›
4116      using "RN[prem]"[where Γ="{«∀x1...∀xn ¬[F]-x1...xn»}"] by blast
4117    AOT_hence ‹Necessary([F])›
4118      using "≡dfI"[OF "contingent-properties:1"] by blast
4119  }
4120  ultimately AOT_have ‹Necessary([F]) ∨ Impossible([F])›
4121    using "∨E"(1) "∨I" "→I" by metis
4122  AOT_thus ‹NonContingent([F])›
4123    using "≡dfI"[OF "contingent-properties:3"] by blast
4124qed
4125
4126AOT_theorem "thm-cont-prop:2": ‹Contingent([F]) ≡ ◇∃x [F]x & ◇∃x ¬[F]x›
4127proof -
4128  AOT_have ‹Contingent([F]) ≡ ¬(Necessary([F]) ∨ Impossible([F]))›
4129    using "contingent-properties:4"[THEN "≡Df", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst]]
4130    by blast
4131  also AOT_have ‹... ≡ ¬Necessary([F]) & ¬Impossible([F])›
4132    using "oth-class-taut:5:d" by fastforce
4133  also AOT_have ‹... ≡ ¬Impossible([F]) & ¬Necessary([F])›
4134    by (simp add: "Commutativity of &")
4135  also AOT_have ‹... ≡ ◇∃x [F]x & ¬Necessary([F])›
4136  proof (rule "oth-class-taut:4:e"[THEN "→E"])
4137    AOT_have ‹¬Impossible([F]) ≡ ¬□¬ ∃x [F]x›
4138      apply (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
4139      apply (AOT_subst "«∃x [F]x»" "«¬ ∀x ¬[F]x»")
4140       apply (simp add: "conventions:4" "≡Df")
4141      apply (AOT_subst_rev "«∀x ¬[F]x»" "«¬¬∀x ¬[F]x»" )
4142       apply (simp add: "oth-class-taut:3:b")
4143      using "contingent-properties:2"[THEN "≡Df", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst]] by blast
4144    also AOT_have ‹... ≡ ◇∃x [F]x›
4145      using "conventions:5"[THEN "≡Df", symmetric] by blast
4146    finally AOT_show ‹¬Impossible([F]) ≡ ◇∃x [F]x› .
4147  qed
4148  also AOT_have ‹... ≡ ◇∃x [F]x & ◇∃x ¬[F]x›
4149  proof (rule "oth-class-taut:4:f"[THEN "→E"])
4150    AOT_have ‹¬Necessary([F]) ≡ ¬□¬∃x ¬[F]x›
4151      apply (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
4152      apply (AOT_subst "«∃x ¬[F]x»" "«¬ ∀x ¬¬[F]x»")
4153       apply (simp add: "conventions:4" "≡Df")
4154      apply (AOT_subst_rev "λ κ . «[F]κ»" "λ κ . «¬¬[F]κ»")
4155       apply (simp add: "oth-class-taut:3:b")
4156      apply (AOT_subst_rev "«∀x [F]x»" "«¬¬∀x [F]x»")
4157      by (auto simp: "oth-class-taut:3:b" "contingent-properties:1" "≡Df")
4158    also AOT_have ‹... ≡ ◇∃x ¬[F]x›
4159      using "conventions:5"[THEN "≡Df", symmetric] by blast
4160    finally AOT_show ‹¬Necessary([F]) ≡ ◇∃x ¬[F]x›.
4161  qed
4162  finally show ?thesis.
4163qed
4164
4165AOT_theorem "thm-cont-prop:3": ‹Contingent([F]) ≡ Contingent([F]-)› for F::‹<κ> AOT_var›
4166proof -
4167  {
4168    fix Π :: ‹<κ>›
4169    AOT_assume ‹Π↓›
4170    moreover AOT_have ‹∀F (Contingent([F]) ≡ ◇∃x [F]x & ◇∃x ¬[F]x)›
4171      using "thm-cont-prop:2" GEN by fast
4172    ultimately  AOT_have ‹Contingent([Π]) ≡ ◇∃x [Π]x & ◇∃x ¬[Π]x›
4173      using "thm-cont-prop:2" "∀E" by fast
4174  } note 1 = this
4175  AOT_have ‹Contingent([F]) ≡ ◇∃x [F]x & ◇∃x ¬[F]x›
4176    using "thm-cont-prop:2" by blast
4177  also AOT_have ‹... ≡ ◇∃x ¬[F]x & ◇∃x [F]x›
4178    by (simp add: "Commutativity of &")
4179  also AOT_have ‹... ≡ ◇∃x [F]-x & ◇∃x [F]x›
4180    by (AOT_subst "λ κ . «[F]-κ»"  "λκ . «¬[F]κ»")
4181       (auto simp: "thm-relation-negation:1" "oth-class-taut:3:a")
4182  also AOT_have ‹... ≡ ◇∃x [F]-x & ◇∃x ¬[F]-x›
4183    by (AOT_subst_rev "λ κ . «¬[F]-κ»"  "λκ . «[F]κ»")
4184       (auto simp: "thm-relation-negation:2" "oth-class-taut:3:a")
4185  also AOT_have ‹... ≡ Contingent([F]-)›
4186    using 1[OF "rel-neg-T:3", symmetric] by blast
4187  finally show ?thesis.
4188qed
4189
4190AOT_define concrete_if_concrete :: ‹Π› ("L")  L_def: ‹L =df [λx E!x → E!x]›
4191
4192AOT_theorem "thm-noncont-e-e:1": ‹Necessary(L)›
4193proof -
4194  AOT_modally_strict {
4195    fix x
4196    AOT_have ‹[λx E!x → E!x]↓› by "cqt:2[lambda]"
4197    moreover AOT_have ‹x↓› using "cqt:2[const_var]"[axiom_inst] by blast
4198    moreover AOT_have ‹E!x → E!x› using "if-p-then-p" by blast
4199    ultimately AOT_have ‹[λx E!x → E!x]x›
4200      using "β←C" by blast
4201  }
4202  AOT_hence 0: ‹□∀x [λx E!x → E!x]x›
4203    using RN GEN by blast
4204  show ?thesis
4205    apply (rule "=dfI"(2)[OF L_def])
4206     apply "cqt:2[lambda]"
4207    by (rule "contingent-properties:1"[THEN "≡dfI", OF 0])
4208qed
4209
4210AOT_theorem "thm-noncont-e-e:2": ‹Impossible([L]-)›
4211proof -
4212  AOT_modally_strict {
4213    fix x
4214
4215    AOT_have 0: ‹∀F (¬[F]-x ≡ [F]x)›
4216      using "thm-relation-negation:2" GEN by fast
4217    AOT_have ‹¬[λx E!x → E!x]-x ≡ [λx E!x → E!x]x›
4218      by (rule 0[THEN "∀E"(1)]) "cqt:2[lambda]"
4219    moreover {
4220      AOT_have ‹[λx E!x → E!x]↓› by "cqt:2[lambda]"
4221      moreover AOT_have ‹x↓› using "cqt:2[const_var]"[axiom_inst] by blast
4222      moreover AOT_have ‹E!x → E!x› using "if-p-then-p" by blast
4223      ultimately AOT_have ‹[λx E!x → E!x]x›
4224        using "β←C" by blast
4225    }
4226    ultimately AOT_have ‹¬[λx E!x → E!x]-x›
4227      using "≡E" by blast
4228  }
4229  AOT_hence 0: ‹□∀x ¬[λx E!x → E!x]-x›
4230    using RN GEN by fast
4231  show ?thesis
4232    apply (rule "=dfI"(2)[OF L_def])
4233     apply "cqt:2[lambda]"
4234    apply (rule "contingent-properties:2"[THEN "≡dfI"]; rule "&I")
4235     using "rel-neg-T:3"
4236     apply blast
4237    using 0
4238    by blast
4239qed
4240
4241AOT_theorem "thm-noncont-e-e:3": ‹NonContingent(L)›
4242  using "thm-noncont-e-e:1"
4243  by (rule "contingent-properties:3"[THEN "≡dfI", OF "∨I"(1)])
4244
4245AOT_theorem "thm-noncont-e-e:4": ‹NonContingent([L]-)›
4246proof -
4247  AOT_have 0: ‹∀F (NonContingent([F]) ≡ NonContingent([F]-))›
4248    using "thm-cont-prop:1" "∀I" by fast
4249  moreover AOT_have 1: ‹L↓›
4250    by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
4251  AOT_show ‹NonContingent([L]-)›
4252    using "∀E"(1)[OF 0, OF 1, THEN "≡E"(1), OF "thm-noncont-e-e:3"] by blast
4253qed
4254
4255AOT_theorem "thm-noncont-e-e:5": ‹∃F ∃G (F ≠ «G::<κ>» & NonContingent([F]) & NonContingent([G]))›
4256proof (rule "∃I")+
4257  {
4258    AOT_have ‹∀F [F] ≠ [F]-› using "thm-relation-negation:5" GEN by fast
4259    moreover AOT_have ‹L↓›
4260      by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
4261    ultimately AOT_have ‹L ≠ [L]-› using "∀E" by blast
4262  }
4263  AOT_thus ‹L ≠ [L]- & NonContingent(L) & NonContingent([L]-)›
4264    using "thm-noncont-e-e:3" "thm-noncont-e-e:4" "&I" by metis
4265next
4266  AOT_show ‹[L]-↓›
4267    using "rel-neg-T:3" by blast
4268next
4269  AOT_show ‹L↓›
4270      by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
4271qed
4272
4273AOT_theorem "lem-cont-e:1": ‹◇∃x ([F]x & ◇¬[F]x) ≡ ◇∃x (¬[F]x & ◇[F]x)›
4274proof -
4275  AOT_have ‹◇∃x ([F]x & ◇¬[F]x) ≡ ∃x ◇([F]x & ◇¬[F]x)›
4276    using "BF◇" "CBF◇" "≡I" by blast
4277  also AOT_have ‹… ≡ ∃x (◇[F]x &  ◇¬[F]x)›
4278    by (AOT_subst ‹λκ. «◇([F]κ & ◇¬[F]κ)»›  ‹λ κ .  «◇[F]κ &  ◇¬[F]κ»›)
4279       (auto simp: "S5Basic:11" "cqt-further:7")
4280  also AOT_have ‹… ≡ ∃x (◇¬[F]x & ◇[F]x)›
4281    by (AOT_subst ‹λκ. «◇¬[F]κ & ◇[F]κ»›  ‹λ κ .  «◇[F]κ & ◇¬[F]κ»›)
4282       (auto simp: "Commutativity of &" "cqt-further:7")
4283  also AOT_have ‹… ≡ ∃x ◇(¬[F]x & ◇[F]x)›
4284    by (AOT_subst ‹λ κ .  «◇(¬[F]κ & ◇[F]κ)»› ‹λκ. «◇¬[F]κ & ◇[F]κ»›)
4285       (auto simp: "S5Basic:11" "oth-class-taut:3:a")
4286  also AOT_have ‹… ≡ ◇∃x (¬[F]x & ◇[F]x)›
4287    using "BF◇" "CBF◇" "≡I" by fast
4288  finally show ?thesis.
4289qed
4290
4291AOT_theorem "lem-cont-e:2": ‹◇∃x ([F]x & ◇¬[F]x) ≡ ◇∃x ([F]-x & ◇¬[F]-x)›
4292proof -
4293  AOT_have ‹◇∃x ([F]x & ◇¬[F]x) ≡ ◇∃x (¬[F]x & ◇[F]x)›
4294    using "lem-cont-e:1".
4295  also AOT_have ‹… ≡ ◇∃x ([F]-x & ◇¬[F]-x)›
4296    apply (AOT_subst "λ κ . «¬[F]-κ»" "λ κ . «[F]κ»")
4297     apply (simp add: "thm-relation-negation:2")
4298    apply (AOT_subst "λ κ . «[F]-κ»" "λ κ . «¬[F]κ»")
4299     apply (simp add: "thm-relation-negation:1")
4300    by (simp add: "oth-class-taut:3:a")
4301  finally show ?thesis.
4302qed
4303
4304AOT_theorem "thm-cont-e:1": ‹◇∃x (E!x & ◇¬E!x)›
4305proof (rule "CBF◇"[THEN "→E"])
4306  AOT_have ‹∃x ◇(E!x & ¬𝒜E!x)› using "qml:4"[axiom_inst] "BF◇"[THEN "→E"] by blast
4307  then AOT_obtain a where ‹◇(E!a & ¬𝒜E!a)› using "∃E"[rotated] by blast
4308  AOT_hence θ: ‹◇E!a & ◇¬𝒜E!a›
4309    using "KBasic2:3"[THEN "→E"] by blast
4310  AOT_have ξ: ‹◇E!a & ◇𝒜¬E!a›
4311    by (AOT_subst  "«𝒜¬E!a»" "«¬𝒜E!a»")
4312       (auto simp: "logic-actual-nec:1"[axiom_inst] θ)
4313  AOT_have ζ: ‹◇E!a & 𝒜¬E!a›
4314    by (AOT_subst "«𝒜¬E!a»" "«◇𝒜¬E!a»")
4315       (auto simp add: "Act-Sub:4" ξ)
4316  AOT_hence ‹◇E!a & ◇¬E!a›
4317    using "&E" "&I" "Act-Sub:3"[THEN "→E"] by blast
4318  AOT_hence ‹◇(E!a & ◇¬E!a)› using "S5Basic:11"[THEN "≡E"(2)] by simp
4319  AOT_thus ‹∃x ◇(E!x & ◇¬E!x)› using "∃I"(2) by fast
4320qed
4321
4322AOT_theorem "thm-cont-e:2": ‹◇∃x (¬E!x & ◇E!x)›
4323proof -
4324  AOT_have ‹∀F (◇∃x ([F]x & ◇¬[F]x) ≡ ◇∃x (¬[F]x & ◇[F]x))›
4325    using "lem-cont-e:1" GEN by fast
4326  AOT_hence ‹(◇∃x (E!x & ◇¬E!x) ≡ ◇∃x (¬E!x & ◇E!x))›
4327    using "∀E"(1) "cqt:2[concrete]"[axiom_inst] by blast
4328  thus ?thesis using "thm-cont-e:1" "≡E" by blast
4329qed
4330
4331AOT_theorem "thm-cont-e:3": ‹◇∃x E!x›
4332proof (rule "CBF◇"[THEN "→E"])
4333  AOT_obtain a where ‹◇(E!a & ◇¬E!a)›
4334    using "∃E"[rotated, OF "thm-cont-e:1"[THEN "BF◇"[THEN "→E"]]] by blast
4335  AOT_hence ‹◇E!a›
4336    using "KBasic2:3"[THEN "→E", THEN "&E"(1)] by blast
4337  AOT_thus ‹∃x ◇E!x› using "∃I" by fast
4338qed
4339
4340AOT_theorem "thm-cont-e:4": ‹◇∃x ¬E!x›
4341proof (rule "CBF◇"[THEN "→E"])
4342  AOT_obtain a where ‹◇(E!a & ◇¬E!a)›
4343    using "∃E"[rotated, OF "thm-cont-e:1"[THEN "BF◇"[THEN "→E"]]] by blast
4344  AOT_hence ‹◇◇¬E!a›
4345    using "KBasic2:3"[THEN "→E", THEN "&E"(2)] by blast
4346  AOT_hence ‹◇¬E!a›
4347    using "4◇"[THEN "→E"] by blast
4348  AOT_thus ‹∃x ◇¬E!x› using "∃I" by fast
4349qed
4350
4351AOT_theorem "thm-cont-e:5": ‹Contingent([E!])›
4352proof -
4353  AOT_have ‹∀F (Contingent([F]) ≡ ◇∃x [F]x & ◇∃x ¬[F]x)›
4354    using "thm-cont-prop:2" GEN by fast
4355  AOT_hence ‹Contingent([E!]) ≡ ◇∃x E!x & ◇∃x ¬E!x›
4356    using "∀E"(1) "cqt:2[concrete]"[axiom_inst] by blast
4357  thus ?thesis
4358    using "thm-cont-e:3" "thm-cont-e:4" "≡E"(2) "&I" by blast
4359qed
4360
4361AOT_theorem "thm-cont-e:6": ‹Contingent([E!]-)›
4362proof -
4363  AOT_have ‹∀F (Contingent([«F::<κ>»]) ≡ Contingent([F]-))›
4364    using "thm-cont-prop:3" GEN by fast
4365  AOT_hence ‹Contingent([E!]) ≡ Contingent([E!]-)›
4366    using "∀E" "cqt:2[concrete]"[axiom_inst] by fast
4367  thus ?thesis using "thm-cont-e:5" "≡E" by blast
4368qed
4369
4370AOT_theorem "thm-cont-e:7": ‹∃F∃G (Contingent([«F::<κ>»]) & Contingent([G]) & F ≠ G)›
4371proof (rule "∃I")+
4372  AOT_have ‹∀F [«F::<κ>»] ≠ [F]-› using "thm-relation-negation:5" GEN by fast
4373  AOT_hence ‹[E!] ≠ [E!]-›
4374    using "∀E" "cqt:2[concrete]"[axiom_inst] by fast
4375  AOT_thus ‹Contingent([E!]) & Contingent([E!]-) & [E!] ≠ [E!]-›
4376    using "thm-cont-e:5" "thm-cont-e:6" "&I" by metis
4377next
4378  AOT_show ‹E!-↓›
4379    by (fact AOT)
4380next
4381  AOT_show ‹E!↓› by (fact "cqt:2[concrete]"[axiom_inst])
4382qed
4383
4384AOT_theorem "property-facts:1": ‹NonContingent([F]) → ¬∃G (Contingent([G]) & G = F)›
4385proof (rule "→I"; rule "raa-cor:2")
4386  AOT_assume ‹NonContingent([F])›
4387  AOT_hence 1: ‹Necessary([F]) ∨ Impossible([F])›
4388    using "contingent-properties:3"[THEN "≡dfE"] by blast
4389  AOT_assume ‹∃G (Contingent([G]) & G = F)›
4390  then AOT_obtain G where ‹Contingent([G]) & G = F› using "∃E"[rotated] by blast
4391  AOT_hence ‹Contingent([F])› using "rule=E" "&E" by blast
4392  AOT_hence ‹¬(Necessary([F]) ∨ Impossible([F]))›
4393    using "contingent-properties:4"[THEN "≡Df", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst], THEN "≡E"(1)] by blast
4394  AOT_thus ‹(Necessary([F]) ∨ Impossible([F])) & ¬(Necessary([F]) ∨ Impossible([F]))›
4395    using 1 "&I" by blast
4396qed
4397
4398AOT_theorem "property-facts:2": ‹Contingent([F]) → ¬∃G (NonContingent([G]) & G = F)›
4399proof (rule "→I"; rule "raa-cor:2")
4400  AOT_assume ‹Contingent([F])›
4401  AOT_hence 1: ‹¬(Necessary([F]) ∨ Impossible([F]))›
4402    using "contingent-properties:4"[THEN "≡Df", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst], THEN "≡E"(1)] by blast
4403  AOT_assume ‹∃G (NonContingent([G]) & G = F)›
4404  then AOT_obtain G where ‹NonContingent([G]) & G = F› using "∃E"[rotated] by blast
4405  AOT_hence ‹NonContingent([F])› using "rule=E" "&E" by blast
4406  AOT_hence ‹Necessary([F]) ∨ Impossible([F])›
4407    using "contingent-properties:3"[THEN "≡dfE"] by blast
4408  AOT_thus ‹(Necessary([F]) ∨ Impossible([F])) & ¬(Necessary([F]) ∨ Impossible([F]))›
4409    using 1 "&I" by blast
4410qed
4411
4412AOT_theorem "property-facts:3": ‹L ≠ [L]- & L ≠ E! & L ≠ E!- & [L]- ≠ [E!]- & E! ≠ [E!]-›
4413proof -
4414  AOT_have noneqI: ‹Π ≠ Π'› if ‹φ{Π}› and ‹¬φ{Π'}› for φ Π Π'
4415    apply (rule "=-infix"[THEN "≡dfI"]; rule "raa-cor:2")
4416    using "rule=E"[where φ=φ and τ=Π and σ = Π'] that "&I" by blast
4417  AOT_have contingent_denotes: ‹Π↓› if ‹Contingent([Π])› for Π
4418    using that "contingent-properties:4"[THEN "≡dfE", THEN "&E"(1)] by blast
4419  AOT_have not_noncontingent_if_contingent: ‹¬NonContingent([Π])› if ‹Contingent([Π])› for Π
4420  proof(rule RAA(2))
4421    AOT_show ‹¬(Necessary([Π]) ∨ Impossible([Π]))›
4422      using that "contingent-properties:4"[THEN "≡Df", THEN "≡S"(1), OF contingent_denotes[OF that], THEN "≡E"(1)] by blast
4423  next
4424    AOT_assume ‹NonContingent([Π])›
4425    AOT_thus ‹Necessary([Π]) ∨ Impossible([Π])›
4426      using "contingent-properties:3"[THEN "≡dfE"] by blast
4427  qed
4428
4429  show ?thesis
4430  proof (rule "&I")+
4431    AOT_show ‹L ≠ [L]-›
4432      apply (rule "=dfI"(2)[OF L_def])
4433       apply "cqt:2[lambda]"
4434      apply (rule "∀E"(1)[where φ="λ Π . «Π ≠ [Π]-»"])
4435       apply (rule GEN) apply (fact AOT)
4436      by "cqt:2[lambda]"
4437  next
4438    AOT_show ‹L ≠ E!›
4439      apply (rule noneqI)
4440      using "thm-noncont-e-e:3" not_noncontingent_if_contingent[OF "thm-cont-e:5"]
4441      by auto
4442  next
4443    AOT_show ‹L ≠ E!-›
4444      apply (rule noneqI)
4445      using "thm-noncont-e-e:3" apply fast
4446      apply (rule not_noncontingent_if_contingent)
4447      apply (rule "∀E"(1)[where φ="λ Π . «Contingent([Π]) ≡ Contingent([Π]-)»", rotated, OF contingent_denotes, THEN "≡E"(1), rotated])
4448      using "thm-cont-prop:3" GEN apply fast
4449      using "thm-cont-e:5" by fast+
4450  next
4451    AOT_show ‹[L]- ≠ E!-›
4452      apply (rule noneqI)
4453      using "thm-noncont-e-e:4" apply fast
4454      apply (rule not_noncontingent_if_contingent)
4455      apply (rule "∀E"(1)[where φ="λ Π . «Contingent([Π]) ≡ Contingent([Π]-)»", rotated, OF contingent_denotes, THEN "≡E"(1), rotated])
4456      using "thm-cont-prop:3" GEN apply fast
4457      using "thm-cont-e:5" by fast+
4458  next
4459    AOT_show ‹E! ≠ E!-›
4460      apply (rule "=dfI"(2)[OF L_def])
4461       apply "cqt:2[lambda]"
4462      apply (rule "∀E"(1)[where φ="λ Π . «Π ≠ [Π]-»"])
4463       apply (rule GEN) apply (fact AOT)
4464      by (fact "cqt:2[concrete]"[axiom_inst])
4465  qed
4466qed
4467
4468AOT_theorem "thm-cont-propos:1": ‹NonContingent0(p) ≡ NonContingent0(((p)-))›
4469proof(rule "≡I"; rule "→I")
4470  AOT_assume ‹NonContingent0(p)›
4471  AOT_hence ‹Necessary0(p) ∨ Impossible0(p)›
4472    using "contingent-properties:3[zero]"[THEN "≡dfE"] by blast
4473  moreover {
4474    AOT_assume ‹Necessary0(p)›
4475    AOT_hence 1: ‹□p› using "contingent-properties:1[zero]"[THEN "≡dfE"] by blast
4476    AOT_have ‹□¬((p)-)›
4477      by (AOT_subst "«¬((p)-)»" "AOT_term_of_var p")
4478         (auto simp add: 1 "thm-relation-negation:4")
4479    AOT_hence ‹Impossible0(((p)-))›
4480      by (rule "contingent-properties:2[zero]"[THEN "≡dfI"])
4481  }
4482  moreover {
4483    AOT_assume ‹Impossible0(p)›
4484    AOT_hence 1: ‹□¬p›
4485      by (rule "contingent-properties:2[zero]"[THEN "≡dfE"])
4486    AOT_have ‹□((p)-)›
4487      by (AOT_subst "«((p)-)»" "«¬p»") 
4488         (auto simp: 1 "thm-relation-negation:3")
4489    AOT_hence ‹Necessary0(((p)-))›
4490      by (rule "contingent-properties:1[zero]"[THEN "≡dfI"])
4491  }
4492  ultimately AOT_have ‹Necessary0(((p)-)) ∨ Impossible0(((p)-))›
4493    using "∨E"(1) "∨I" "→I" by metis
4494  AOT_thus ‹NonContingent0(((p)-))›
4495    using "contingent-properties:3[zero]"[THEN "≡dfI"] by blast
4496next
4497  AOT_assume ‹NonContingent0(((p)-))›
4498  AOT_hence ‹Necessary0(((p)-)) ∨ Impossible0(((p)-))›
4499    using "contingent-properties:3[zero]"[THEN "≡dfE"] by blast
4500  moreover {
4501    AOT_assume ‹Impossible0(((p)-))›
4502    AOT_hence 1: ‹□¬((p)-)›
4503      by (rule "contingent-properties:2[zero]"[THEN "≡dfE"])
4504    AOT_have ‹□p›
4505      by (AOT_subst_rev "«¬((p)-)»" "AOT_term_of_var p")
4506         (auto simp: 1 "thm-relation-negation:4")
4507    AOT_hence ‹Necessary0(p)›
4508      using "contingent-properties:1[zero]"[THEN "≡dfI"] by blast
4509  }
4510  moreover {
4511    AOT_assume ‹Necessary0(((p)-))›
4512    AOT_hence 1: ‹□((p)-)›
4513      by (rule "contingent-properties:1[zero]"[THEN "≡dfE"])
4514    AOT_have ‹□¬p›
4515      by (AOT_subst_rev "«((p)-)»" "«¬p»")
4516         (auto simp: 1 "thm-relation-negation:3")
4517    AOT_hence ‹Impossible0(p)›
4518      by (rule "contingent-properties:2[zero]"[THEN "≡dfI"])
4519  }
4520  ultimately AOT_have ‹Necessary0(p) ∨ Impossible0(p)›
4521    using "∨E"(1) "∨I" "→I" by metis
4522  AOT_thus ‹NonContingent0(p)›
4523    using "contingent-properties:3[zero]"[THEN "≡dfI"] by blast
4524qed
4525
4526AOT_theorem "thm-cont-propos:2": ‹Contingent0(φ) ≡ ◇φ & ◇¬φ›
4527proof -
4528  AOT_have ‹Contingent0(φ) ≡ ¬(Necessary0(φ) ∨ Impossible0(φ))›
4529    using "contingent-properties:4[zero]"[THEN "≡Df"] by simp
4530  also AOT_have ‹… ≡ ¬Necessary0(φ) & ¬Impossible0(φ)›
4531    by (fact AOT)
4532  also AOT_have ‹… ≡ ¬Impossible0(φ) & ¬Necessary0(φ)›
4533    by (fact AOT)
4534  also AOT_have ‹… ≡ ◇φ & ◇¬φ›
4535    apply (AOT_subst "«◇φ»" "«¬□¬φ»")
4536     apply (simp add: "conventions:5" "≡Df")
4537    apply (AOT_subst "«Impossible0(φ)»" "«□¬φ»")
4538     apply (simp add: "contingent-properties:2[zero]" "≡Df")
4539    apply (AOT_subst_rev "«¬□φ»" "«◇¬φ»")
4540     apply (simp add: "KBasic:11")
4541    apply (AOT_subst "«Necessary0(φ)»" "«□φ»")
4542     apply (simp add: "contingent-properties:1[zero]" "≡Df")
4543    by (simp add: "oth-class-taut:3:a")
4544  finally show ?thesis.
4545qed
4546
4547AOT_theorem "thm-cont-propos:3": ‹Contingent0(p) ≡ Contingent0(((p)-))›
4548proof -
4549  AOT_have ‹Contingent0(p) ≡ ◇p & ◇¬p› using "thm-cont-propos:2".
4550  also AOT_have ‹… ≡ ◇¬p & ◇p› by (fact AOT)
4551  also AOT_have ‹… ≡ ◇((p)-) & ◇p›
4552    by (AOT_subst "«((p)-)»" "«¬p»")
4553       (auto simp: "thm-relation-negation:3" "oth-class-taut:3:a")
4554  also AOT_have ‹… ≡ ◇((p)-) & ◇¬((p)-)›
4555    by (AOT_subst "«¬((p)-)»" "AOT_term_of_var p")
4556       (auto simp: "thm-relation-negation:4" "oth-class-taut:3:a")
4557  also AOT_have ‹… ≡ Contingent0(((p)-))›
4558    using "thm-cont-propos:2"[symmetric] by blast
4559  finally show ?thesis.
4560qed
4561
4562AOT_define noncontingent_prop :: ‹φ› ("p0")
4563  p0_def: "(p0) =df (∀x (E!x → E!x))"
4564
4565AOT_theorem "thm-noncont-propos:1":  ‹Necessary0((p0))›
4566proof(rule "contingent-properties:1[zero]"[THEN "≡dfI"])
4567  AOT_show ‹□(p0)›
4568    apply (rule "=dfI"(2)[OF p0_def])
4569    using "log-prop-prop:2" apply simp
4570    using "if-p-then-p" RN GEN by fast
4571qed
4572
4573AOT_theorem "thm-noncont-propos:2": ‹Impossible0(((p0)-))›
4574proof(rule "contingent-properties:2[zero]"[THEN "≡dfI"])
4575  AOT_show ‹□¬((p0)-)›
4576    apply (AOT_subst "«((p0)-)»" "«¬p0»")
4577    using "thm-relation-negation:3" GEN "∀E"(1)[rotated, OF "log-prop-prop:2"] apply fast
4578    apply (AOT_subst_rev "«p0»" "«¬¬p0»" )
4579     apply (simp add: "oth-class-taut:3:b")
4580    apply (rule "=dfI"(2)[OF p0_def])
4581    using "log-prop-prop:2" apply simp
4582    using "if-p-then-p" RN GEN by fast
4583qed
4584
4585AOT_theorem "thm-noncont-propos:3": ‹NonContingent0((p0))›
4586  apply(rule "contingent-properties:3[zero]"[THEN "≡dfI"])
4587  using "thm-noncont-propos:1" "∨I" by blast
4588
4589AOT_theorem "thm-noncont-propos:4": ‹NonContingent0(((p0)-))›
4590  apply(rule "contingent-properties:3[zero]"[THEN "≡dfI"])
4591  using "thm-noncont-propos:2" "∨I" by blast
4592
4593AOT_theorem "thm-noncont-propos:5": ‹∃p∃q (NonContingent0((p)) & NonContingent0((q)) & p ≠ q)›
4594proof(rule "∃I")+
4595  AOT_have 0: ‹φ ≠ (φ)-› for φ
4596    using "thm-relation-negation:6" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"] by fast
4597  AOT_thus ‹NonContingent0((p0)) & NonContingent0(((p0)-)) & (p0) ≠ (p0)-›
4598    using "thm-noncont-propos:3" "thm-noncont-propos:4" "&I" by auto
4599qed(auto simp: "log-prop-prop:2")
4600
4601AOT_act_theorem "no-cnac": ‹¬∃x(E!x & ¬𝒜E!x)›
4602proof(rule "raa-cor:2")
4603  AOT_assume ‹∃x(E!x & ¬𝒜E!x)›
4604  then AOT_obtain a where a: ‹E!a & ¬𝒜E!a›
4605    using "∃E"[rotated] by blast
4606  AOT_hence ‹𝒜¬E!a› using "&E" "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] by blast
4607  AOT_hence ‹¬E!a› using "logic-actual"[act_axiom_inst, THEN "→E"] by blast
4608  AOT_hence ‹E!a & ¬E!a› using a "&E" "&I" by blast
4609  AOT_thus ‹p & ¬p› for p using "raa-cor:1" by blast
4610qed
4611
4612AOT_theorem "pos-not-pna:1": ‹¬𝒜∃x (E!x & ¬𝒜E!x)›
4613proof(rule "raa-cor:2")
4614  AOT_assume ‹𝒜∃x (E!x & ¬𝒜E!x)›
4615  AOT_hence ‹∃x 𝒜(E!x & ¬𝒜E!x)›
4616    using "Act-Basic:10"[THEN "≡E"(1)] by blast
4617  then AOT_obtain a where ‹𝒜(E!a & ¬𝒜E!a)› using "∃E"[rotated] by blast
4618  AOT_hence 1: ‹𝒜E!a & 𝒜¬𝒜E!a› using "Act-Basic:2"[THEN "≡E"(1)] by blast
4619  AOT_hence ‹¬𝒜𝒜E!a› using "&E"(2) "logic-actual-nec:1"[axiom_inst, THEN "≡E"(1)] by blast
4620  AOT_hence ‹¬𝒜E!a› using "logic-actual-nec:4"[axiom_inst, THEN "≡E"(1)] RAA by blast
4621  AOT_thus ‹p & ¬p› for p using 1[THEN "&E"(1)] "&I" "raa-cor:1" by blast
4622qed
4623
4624AOT_theorem "pos-not-pna:2": ‹◇¬∃x(E!x & ¬𝒜E!x)›
4625proof (rule RAA(1))
4626  AOT_show ‹¬𝒜∃x (E!x & ¬𝒜E!x)› using "pos-not-pna:1" by blast
4627next
4628  AOT_assume ‹¬◇¬∃x (E!x & ¬𝒜E!x)›
4629  AOT_hence ‹□∃x (E!x & ¬𝒜E!x)›
4630    using "KBasic:12"[THEN "≡E"(2)] by blast
4631  AOT_thus ‹𝒜∃x (E!x & ¬𝒜E!x)›
4632    using "nec-imp-act"[THEN "→E"] by blast
4633qed
4634
4635AOT_theorem "pos-not-pna:3": ‹∃x (◇E!x & ¬𝒜E!x)›
4636proof -
4637  AOT_obtain a where ‹◇(E!a & ¬𝒜E!a)›
4638    using "qml:4"[axiom_inst] "BF◇"[THEN "→E"] "∃E"[rotated] by blast
4639  AOT_hence θ: ‹◇E!a› and ξ: ‹◇¬𝒜E!a› using "KBasic2:3"[THEN "→E"] "&E" by blast+
4640  AOT_have ‹¬□𝒜E!a› using ξ "KBasic:11"[THEN "≡E"(2)] by blast
4641  AOT_hence ‹¬𝒜E!a› using "Act-Basic:6"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)] by blast
4642  AOT_hence ‹◇E!a & ¬𝒜E!a› using θ "&I" by blast
4643  thus ?thesis using "∃I" by fast
4644qed
4645
4646AOT_define contingent_prop :: φ ("q0")
4647  q0_def: ‹(q0) =df (∃x (E!x & ¬𝒜E!x))›
4648
4649AOT_theorem q0_prop: ‹◇q0 & ◇¬q0›
4650  apply (rule "=dfI"(2)[OF q0_def])
4651  apply (fact "log-prop-prop:2")
4652  apply (rule "&I")
4653   apply (fact "qml:4"[axiom_inst])
4654  by (fact "pos-not-pna:2")
4655
4656AOT_theorem "basic-prop:1": ‹Contingent0((q0))›
4657proof(rule "contingent-properties:4[zero]"[THEN "≡dfI"])
4658  AOT_have ‹¬Necessary0((q0)) & ¬Impossible0((q0))›
4659  proof (rule "&I"; rule "=dfI"(2)[OF q0_def]; (rule "log-prop-prop:2" | rule "raa-cor:2"))
4660    AOT_assume ‹Necessary0(∃x (E!x & ¬𝒜E!x))›
4661    AOT_hence ‹□∃x (E!x & ¬𝒜E!x)›
4662      using "contingent-properties:1[zero]"[THEN "≡dfE"] by blast
4663    AOT_hence ‹𝒜∃x (E!x & ¬𝒜E!x)›
4664      using "Act-Basic:8"[THEN "→E"] "qml:2"[axiom_inst, THEN "→E"] by blast
4665    AOT_thus ‹𝒜∃x (E!x & ¬𝒜E!x) & ¬𝒜∃x (E!x & ¬𝒜E!x)›
4666      using "pos-not-pna:1" "&I" by blast
4667  next
4668    AOT_assume ‹Impossible0(∃x (E!x & ¬𝒜E!x))›
4669    AOT_hence ‹□¬(∃x (E!x & ¬𝒜E!x))›
4670      using "contingent-properties:2[zero]"[THEN "≡dfE"] by blast
4671    AOT_hence ‹¬◇(∃x (E!x & ¬𝒜E!x))› using "KBasic2:1"[THEN "≡E"(1)] by blast
4672    AOT_thus ‹◇(∃x (E!x & ¬𝒜E!x)) & ¬◇(∃x (E!x & ¬𝒜E!x))›
4673      using "qml:4"[axiom_inst] "&I" by blast
4674  qed
4675  AOT_thus ‹¬(Necessary0((q0)) ∨ Impossible0((q0)))›
4676    using "oth-class-taut:5:d" "≡E"(2) by blast
4677qed
4678
4679AOT_theorem "basic-prop:2": ‹∃p Contingent0((p))›
4680  using "∃I"(1)[rotated, OF "log-prop-prop:2"] "basic-prop:1" by blast
4681
4682AOT_theorem "basic-prop:3": ‹Contingent0(((q0)-))›
4683  apply (AOT_subst "«(q0)-»" "«¬q0»")
4684   apply (insert "thm-relation-negation:3" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"]; fast)
4685  apply (rule "contingent-properties:4[zero]"[THEN "≡dfI"])
4686  apply (rule "oth-class-taut:5:d"[THEN "≡E"(2)])
4687  apply (rule "&I")
4688   apply (rule "contingent-properties:1[zero]"[THEN "df-rules-formulas[3]", THEN "useful-tautologies:5"[THEN "→E"], THEN "→E"])
4689   apply (rule "conventions:5"[THEN "≡dfE"])
4690   apply (rule "=dfE"(2)[OF q0_def])
4691    apply (rule "log-prop-prop:2")
4692   apply (rule q0_prop[THEN "&E"(1)])
4693  apply (rule "contingent-properties:2[zero]"[THEN "df-rules-formulas[3]", THEN "useful-tautologies:5"[THEN "→E"], THEN "→E"])
4694  apply (rule "conventions:5"[THEN "≡dfE"])
4695  by (rule q0_prop[THEN "&E"(2)])
4696
4697AOT_theorem "basic-prop:4": ‹∃p∃q (p ≠ q & Contingent0(p) & Contingent0(q))›
4698proof(rule "∃I")+
4699  AOT_have 0: ‹φ ≠ (φ)-› for φ
4700    using "thm-relation-negation:6" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"] by fast
4701  AOT_show ‹(q0) ≠ (q0)- & Contingent0(q0) & Contingent0(((q0)-))›
4702    using "basic-prop:1" "basic-prop:3" "&I" 0 by presburger
4703qed(auto simp: "log-prop-prop:2")
4704
4705AOT_theorem "proposition-facts:1": ‹NonContingent0(p) → ¬∃q (Contingent0(q) & q = p)›
4706proof(rule "→I"; rule "raa-cor:2")
4707  AOT_assume ‹NonContingent0(p)›
4708  AOT_hence 1: ‹Necessary0(p) ∨ Impossible0(p)›
4709    using "contingent-properties:3[zero]"[THEN "≡dfE"] by blast
4710  AOT_assume ‹∃q (Contingent0(q) & q = p)›
4711  then AOT_obtain q where ‹Contingent0(q) & q = p› using "∃E"[rotated] by blast
4712  AOT_hence ‹Contingent0(p)› using "rule=E" "&E" by fast
4713  AOT_thus ‹(Necessary0(p) ∨ Impossible0(p)) & ¬(Necessary0(p) ∨ Impossible0(p))›
4714    using "contingent-properties:4[zero]"[THEN "≡dfE"] 1 "&I" by blast
4715qed
4716
4717AOT_theorem "proposition-facts:2": ‹Contingent0(p) → ¬∃q (NonContingent0(q) & q = p)›
4718proof(rule "→I"; rule "raa-cor:2")
4719  AOT_assume ‹Contingent0(p)›
4720  AOT_hence 1: ‹¬(Necessary0(p) ∨ Impossible0(p))›
4721    using "contingent-properties:4[zero]"[THEN "≡dfE"] by blast
4722  AOT_assume ‹∃q (NonContingent0(q) & q = p)›
4723  then AOT_obtain q where ‹NonContingent0(q) & q = p› using "∃E"[rotated] by blast
4724  AOT_hence ‹NonContingent0(p)› using "rule=E" "&E" by fast
4725  AOT_thus ‹(Necessary0(p) ∨ Impossible0(p)) & ¬(Necessary0(p) ∨ Impossible0(p))›
4726    using "contingent-properties:3[zero]"[THEN "≡dfE"] 1 "&I" by blast
4727qed
4728
4729AOT_theorem "proposition-facts:3": ‹(p0) ≠ (p0)- & (p0) ≠ (q0) & (p0) ≠ (q0)- & (p0)- ≠ (q0)- & (q0) ≠ (q0)-›
4730proof -
4731  {
4732    fix χ φ ψ
4733    AOT_assume ‹χ{φ}›
4734    moreover AOT_assume ‹¬χ{ψ}›
4735    ultimately AOT_have ‹¬(χ{φ} ≡ χ{ψ})›
4736      using RAA "≡E" by metis
4737    moreover {
4738      AOT_have ‹∀p∀q ((¬(χ{p} ≡ χ{q})) → p ≠ q)›
4739        by (rule "∀I"; rule "∀I"; rule "pos-not-equiv-ne:4[zero]")
4740      AOT_hence ‹((¬(χ{φ} ≡ χ{ψ})) → φ ≠ ψ)›
4741        using "∀E" "log-prop-prop:2" by blast
4742    }
4743    ultimately AOT_have ‹φ ≠ ψ›
4744      using "→E" by blast
4745  } note 0 = this
4746  AOT_have contingent_neg: ‹Contingent0(φ) ≡ Contingent0(((φ)-))› for φ
4747    using "thm-cont-propos:3" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"] by fast
4748  AOT_have not_noncontingent_if_contingent: ‹¬NonContingent0(φ)› if ‹Contingent0(φ)› for φ
4749    apply (rule "contingent-properties:3[zero]"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
4750    using that "contingent-properties:4[zero]"[THEN "≡dfE"] by blast
4751  show ?thesis
4752    apply (rule "&I")+
4753    using "thm-relation-negation:6" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"] apply fast
4754       apply (rule 0)
4755    using "thm-noncont-propos:3" apply fast
4756       apply (rule not_noncontingent_if_contingent)
4757       apply (fact AOT)
4758      apply (rule 0)
4759    apply (rule "thm-noncont-propos:3")
4760      apply (rule not_noncontingent_if_contingent)
4761      apply (rule contingent_neg[THEN "≡E"(1)])
4762      apply (fact AOT)
4763     apply (rule 0)
4764    apply (rule "thm-noncont-propos:4")
4765      apply (rule not_noncontingent_if_contingent)
4766      apply (rule contingent_neg[THEN "≡E"(1)])
4767     apply (fact AOT)
4768    using "thm-relation-negation:6" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"] by fast
4769qed
4770
4771AOT_define ContingentlyTrue :: ‹φ ⇒ φ› ("ContingentlyTrue'(_')")
4772  "cont-tf:1": ‹ContingentlyTrue(p) ≡df p & ◇¬p›
4773
4774AOT_define ContingentlyFalse :: ‹φ ⇒ φ› ("ContingentlyFalse'(_')")
4775  "cont-tf:2": ‹ContingentlyFalse(p) ≡df ¬p & ◇p›
4776
4777AOT_theorem "cont-true-cont:1": ‹ContingentlyTrue((p)) → Contingent0((p))›
4778proof(rule "→I")
4779  AOT_assume ‹ContingentlyTrue((p))›
4780  AOT_hence 1: ‹p› and 2: ‹◇¬p› using "cont-tf:1"[THEN "≡dfE"] "&E" by blast+
4781  AOT_have ‹¬Necessary0((p))›
4782    apply (rule "contingent-properties:1[zero]"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
4783    using 2 "KBasic:11"[THEN "≡E"(2)] by blast
4784  moreover AOT_have ‹¬Impossible0((p))›
4785    apply (rule "contingent-properties:2[zero]"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
4786    apply (rule "conventions:5"[THEN "≡dfE"])
4787    using "T◇"[THEN "→E", OF 1].
4788  ultimately AOT_have ‹¬(Necessary0((p)) ∨ Impossible0((p)))›
4789    using DeMorgan(2)[THEN "≡E"(2)] "&I" by blast
4790  AOT_thus ‹Contingent0((p))›
4791    using "contingent-properties:4[zero]"[THEN "≡dfI"] by blast
4792qed
4793
4794AOT_theorem "cont-true-cont:2": ‹ContingentlyFalse((p)) → Contingent0((p))›
4795proof(rule "→I")
4796  AOT_assume ‹ContingentlyFalse((p))›
4797  AOT_hence 1: ‹¬p› and 2: ‹◇p› using "cont-tf:2"[THEN "≡dfE"] "&E" by blast+
4798  AOT_have ‹¬Necessary0((p))›
4799    apply (rule "contingent-properties:1[zero]"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
4800    using "KBasic:11"[THEN "≡E"(2)] "T◇"[THEN "→E", OF 1] by blast
4801  moreover AOT_have ‹¬Impossible0((p))›
4802    apply (rule "contingent-properties:2[zero]"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
4803    apply (rule "conventions:5"[THEN "≡dfE"])
4804    using 2.
4805  ultimately AOT_have ‹¬(Necessary0((p)) ∨ Impossible0((p)))›
4806    using DeMorgan(2)[THEN "≡E"(2)] "&I" by blast
4807  AOT_thus ‹Contingent0((p))›
4808    using "contingent-properties:4[zero]"[THEN "≡dfI"] by blast
4809qed
4810
4811AOT_theorem "cont-true-cont:3": ‹ContingentlyTrue((p)) ≡ ContingentlyFalse(((p)-))›
4812proof(rule "≡I"; rule "→I")
4813  AOT_assume ‹ContingentlyTrue((p))›
4814  AOT_hence 0: ‹p & ◇¬p› using "cont-tf:1"[THEN "≡dfE"] by blast
4815  AOT_have 1: ‹ContingentlyFalse(¬p)›
4816    apply (rule "cont-tf:2"[THEN "≡dfI"])
4817    apply (AOT_subst_rev "AOT_term_of_var p" "«¬¬p»")
4818    by (auto simp: "oth-class-taut:3:b" 0)
4819  AOT_show ‹ContingentlyFalse(((p)-))›
4820    apply (AOT_subst "«(p)-»" "«¬p»")
4821    by (auto simp: "thm-relation-negation:3" 1)
4822next
4823  AOT_assume 1: ‹ContingentlyFalse(((p)-))›
4824  AOT_have ‹ContingentlyFalse(¬p)›
4825    by (AOT_subst_rev "«(p)-»" "«¬p»")
4826       (auto simp: "thm-relation-negation:3" 1)
4827  AOT_hence ‹¬¬p & ◇¬p› using "cont-tf:2"[THEN "≡dfE"] by blast
4828  AOT_hence ‹p & ◇¬p›
4829    using "&I" "&E" "useful-tautologies:1"[THEN "→E"] by metis
4830  AOT_thus ‹ContingentlyTrue((p))›
4831    using "cont-tf:1"[THEN "≡dfI"] by blast
4832qed
4833
4834AOT_theorem "cont-true-cont:4": ‹ContingentlyFalse((p)) ≡ ContingentlyTrue(((p)-))›
4835proof(rule "≡I"; rule "→I")
4836  AOT_assume ‹ContingentlyFalse(p)›
4837  AOT_hence 0: ‹¬p & ◇p›
4838    using "cont-tf:2"[THEN "≡dfE"] by blast
4839  AOT_have ‹¬p & ◇¬¬p›
4840    by (AOT_subst_rev "AOT_term_of_var p" "«¬¬p»")
4841       (auto simp: "oth-class-taut:3:b" 0)
4842  AOT_hence 1: ‹ContingentlyTrue(¬p)›
4843    by (rule "cont-tf:1"[THEN "≡dfI"])
4844  AOT_show ‹ContingentlyTrue(((p)-))›
4845    by (AOT_subst "«(p)-»" "«¬p»")
4846       (auto simp: "thm-relation-negation:3" 1)
4847next
4848  AOT_assume 1: ‹ContingentlyTrue(((p)-))›
4849  AOT_have ‹ContingentlyTrue(¬p)›
4850    by (AOT_subst_rev "«(p)-»" "«¬p»")
4851       (auto simp add: "thm-relation-negation:3" 1)
4852  AOT_hence 2: ‹¬p & ◇¬¬p› using "cont-tf:1"[THEN "≡dfE"] by blast
4853  AOT_have ‹◇p›
4854    by (AOT_subst "AOT_term_of_var p" "«¬¬p»")
4855       (auto simp add: "oth-class-taut:3:b" 2[THEN "&E"(2)])
4856  AOT_hence ‹¬p & ◇p› using 2[THEN "&E"(1)] "&I" by blast
4857  AOT_thus ‹ContingentlyFalse(p)›
4858    by (rule "cont-tf:2"[THEN "≡dfI"])
4859qed
4860
4861AOT_theorem "cont-true-cont:5": ‹(ContingentlyTrue((p)) & Necessary0((q))) → p ≠ q›
4862proof (rule "→I"; frule "&E"(1); drule "&E"(2); rule "raa-cor:1")
4863  AOT_assume ‹ContingentlyTrue((p))›
4864  AOT_hence ‹◇¬p›
4865    using "cont-tf:1"[THEN "≡dfE"] "&E" by blast
4866  AOT_hence 0: ‹¬□p› using "KBasic:11"[THEN "≡E"(2)] by blast
4867  AOT_assume ‹Necessary0((q))›
4868  moreover AOT_assume ‹¬(p ≠ q)›
4869  AOT_hence ‹p = q›
4870    using "=-infix"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1)]
4871          "useful-tautologies:1"[THEN "→E"] by blast
4872  ultimately AOT_have ‹Necessary0((p))› using "rule=E" id_sym by blast
4873  AOT_hence ‹□p›
4874    using "contingent-properties:1[zero]"[THEN "≡dfE"] by blast
4875  AOT_thus ‹□p & ¬□p› using 0 "&I" by blast
4876qed
4877
4878AOT_theorem "cont-true-cont:6": ‹(ContingentlyFalse((p)) & Impossible0((q))) → p ≠ q›
4879proof (rule "→I"; frule "&E"(1); drule "&E"(2); rule "raa-cor:1")
4880  AOT_assume ‹ContingentlyFalse((p))›
4881  AOT_hence ‹◇p›
4882    using "cont-tf:2"[THEN "≡dfE"] "&E" by blast
4883  AOT_hence 1: ‹¬□¬p›
4884    using "conventions:5"[THEN "≡dfE"] by blast
4885  AOT_assume ‹Impossible0((q))›
4886  moreover AOT_assume ‹¬(p ≠ q)›
4887  AOT_hence ‹p = q›
4888    using "=-infix"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1)]
4889          "useful-tautologies:1"[THEN "→E"] by blast
4890  ultimately AOT_have ‹Impossible0((p))› using "rule=E" id_sym by blast
4891  AOT_hence ‹□¬p›
4892    using "contingent-properties:2[zero]"[THEN "≡dfE"] by blast
4893  AOT_thus ‹□¬p & ¬□¬p› using 1 "&I" by blast
4894qed
4895
4896AOT_act_theorem "q0cf:1": ‹ContingentlyFalse(q0)›
4897  apply (rule "cont-tf:2"[THEN "≡dfI"])
4898  apply (rule "=dfI"(2)[OF q0_def])
4899   apply (fact "log-prop-prop:2")
4900  apply (rule "&I")
4901   apply (fact "no-cnac")
4902  by (fact "qml:4"[axiom_inst])
4903
4904AOT_act_theorem "q0cf:2": ‹ContingentlyTrue(((q0)-))›
4905  apply (rule "cont-tf:1"[THEN "≡dfI"])
4906  apply (rule "=dfI"(2)[OF q0_def])
4907   apply (fact "log-prop-prop:2")
4908  apply (rule "&I")
4909     apply (rule "thm-relation-negation:3"[unvarify p, OF "log-prop-prop:2", THEN "≡E"(2)])
4910     apply (fact "no-cnac")
4911    apply (rule "rule=E"[rotated, OF "thm-relation-negation:7"[unvarify p, OF "log-prop-prop:2", THEN id_sym]])
4912  apply (AOT_subst_rev "«∃x (E!x & ¬𝒜E!x)»" "«¬¬(∃x  (E!x & ¬𝒜E!x))»")
4913  by (auto simp: "oth-class-taut:3:b" "qml:4"[axiom_inst])
4914
4915(* TODO: q0cf-rem skipped for now *)
4916
4917AOT_theorem "cont-tf-thm:1": ‹∃p ContingentlyTrue((p))›
4918proof(rule "∨E"(1)[OF "exc-mid"]; rule "→I"; rule "∃I")
4919  AOT_assume ‹q0›
4920  AOT_hence ‹q0 & ◇¬q0› using q0_prop[THEN "&E"(2)] "&I" by blast
4921  AOT_thus ‹ContingentlyTrue(q0)›
4922    by (rule "cont-tf:1"[THEN "≡dfI"])
4923next
4924  AOT_assume ‹¬q0›
4925  AOT_hence ‹¬q0 & ◇q0› using q0_prop[THEN "&E"(1)] "&I" by blast
4926  AOT_hence ‹ContingentlyFalse(q0)›
4927    by (rule "cont-tf:2"[THEN "≡dfI"])
4928  AOT_thus ‹ContingentlyTrue(((q0)-))›
4929    by (rule "cont-true-cont:4"[unvarify p, OF "log-prop-prop:2", THEN "≡E"(1)])
4930qed(auto simp: "log-prop-prop:2")
4931
4932
4933AOT_theorem "cont-tf-thm:2": ‹∃p ContingentlyFalse((p))›
4934proof(rule "∨E"(1)[OF "exc-mid"]; rule "→I"; rule "∃I")
4935  AOT_assume ‹q0›
4936  AOT_hence ‹q0 & ◇¬q0› using q0_prop[THEN "&E"(2)] "&I" by blast
4937  AOT_hence ‹ContingentlyTrue(q0)›
4938    by (rule "cont-tf:1"[THEN "≡dfI"])
4939  AOT_thus ‹ContingentlyFalse(((q0)-))›
4940    by (rule "cont-true-cont:3"[unvarify p, OF "log-prop-prop:2", THEN "≡E"(1)])
4941next
4942  AOT_assume ‹¬q0›
4943  AOT_hence ‹¬q0 & ◇q0› using q0_prop[THEN "&E"(1)] "&I" by blast
4944  AOT_thus ‹ContingentlyFalse(q0)›
4945    by (rule "cont-tf:2"[THEN "≡dfI"])
4946qed(auto simp: "log-prop-prop:2")
4947
4948(* TODO: inspect modally strict subproof involving obtained variable *)
4949AOT_theorem "property-facts1:1": ‹∃F∃x ([F]x & ◇¬[F]x)›
4950proof -
4951  fix x
4952  AOT_obtain p1 where ‹ContingentlyTrue((p1))›
4953    using "cont-tf-thm:1" "∃E"[rotated] by blast
4954  AOT_hence 1: ‹p1 & ◇¬p1› using "cont-tf:1"[THEN "≡dfE"] by blast
4955  AOT_modally_strict {
4956    AOT_have ‹for arbitrary p: ⊢□ ([λz p]x ≡ p)›
4957      by (rule "beta-C-cor:3"[THEN "∀E"(2)]) cqt_2_lambda_inst_prover
4958    AOT_hence ‹for arbitrary p: ⊢□ □ ([λz p]x ≡ p)›
4959      by (rule RN)
4960    AOT_hence ‹∀p □([λz p]x ≡ p)› using GEN by fast
4961    AOT_hence ‹□([λz p1]x ≡ p1)› using "∀E" by fast
4962  } note 2 = this
4963  AOT_hence ‹□([λz p1]x ≡ p1)› using "∀E" by blast
4964  AOT_hence ‹[λz p1]x› using 1[THEN "&E"(1)] "qml:2"[axiom_inst, THEN "→E"] "≡E"(2) by blast
4965  moreover AOT_have ‹◇¬[λz p1]x›
4966    apply (AOT_subst_using subst: 2[THEN "qml:2"[axiom_inst, THEN "→E"]])
4967    using 1[THEN "&E"(2)] by blast
4968  ultimately AOT_have ‹[λz p1]x & ◇¬[λz p1]x› using "&I" by blast
4969  AOT_hence ‹∃x ([λz p1]x & ◇¬[λz p1]x)› using "∃I"(2) by fast
4970  moreover AOT_have ‹[λz p1]↓› by "cqt:2[lambda]"
4971  ultimately AOT_show ‹∃F∃x ([F]x & ◇¬[F]x)› by (rule "∃I"(1))
4972qed
4973
4974(* TODO: inspect modally strict subproof involving obtained variable *)
4975AOT_theorem "property-facts1:2": ‹∃F∃x (¬[F]x & ◇[F]x)›
4976proof -
4977  fix x
4978  AOT_obtain p1 where ‹ContingentlyFalse((p1))›
4979    using "cont-tf-thm:2" "∃E"[rotated] by blast
4980  AOT_hence 1: ‹¬p1 & ◇p1› using "cont-tf:2"[THEN "≡dfE"] by blast
4981  AOT_modally_strict {
4982    AOT_have ‹for arbitrary p: ⊢□ ([λz p]x ≡ p)›
4983      by (rule "beta-C-cor:3"[THEN "∀E"(2)]) cqt_2_lambda_inst_prover
4984    AOT_hence ‹for arbitrary p: ⊢□ (¬[λz p]x ≡ ¬p)›
4985      using "oth-class-taut:4:b" "≡E" by blast
4986    AOT_hence ‹for arbitrary p: ⊢□ □(¬[λz p]x ≡ ¬p)›
4987      by (rule RN)
4988    AOT_hence ‹∀p □(¬[λz p]x ≡ ¬p)› using GEN by fast
4989    AOT_hence ‹□(¬[λz p1]x ≡ ¬p1)› using "∀E" by fast
4990  } note 2 = this
4991  AOT_hence ‹□(¬[λz p1]x ≡ ¬p1)› using "∀E" by blast
4992  AOT_hence 3: ‹¬[λz p1]x› using 1[THEN "&E"(1)] "qml:2"[axiom_inst, THEN "→E"] "≡E"(2) by blast
4993  AOT_modally_strict {
4994    AOT_have ‹for arbitrary p: ⊢□ ([λz p]x ≡ p)›
4995      by (rule "beta-C-cor:3"[THEN "∀E"(2)]) cqt_2_lambda_inst_prover
4996    AOT_hence ‹for arbitrary p: ⊢□ □([λz p]x ≡ p)›
4997      by (rule RN)
4998    AOT_hence ‹∀p □([λz p]x ≡ p)› using GEN by fast
4999    AOT_hence ‹□([λz p1]x ≡ p1)› using "∀E" by fast
5000  } note 4 = this
5001  AOT_have ‹◇[λz p1]x›
5002    apply (AOT_subst_using subst: 4[THEN "qml:2"[axiom_inst, THEN "→E"]])
5003    using 1[THEN "&E"(2)] by blast
5004  AOT_hence ‹¬[λz p1]x & ◇[λz p1]x› using 3 "&I" by blast
5005  AOT_hence ‹∃x (¬[λz p1]x & ◇[λz p1]x)› using "∃I"(2) by fast
5006  moreover AOT_have ‹[λz p1]↓› by "cqt:2[lambda]"
5007  ultimately AOT_show ‹∃F∃x (¬[F]x & ◇[F]x)› by (rule "∃I"(1))
5008qed
5009
5010context
5011begin
5012
5013private AOT_lemma eqnotnec_123_Aux_ζ: ‹[L]x ≡ (E!x → E!x)›
5014    apply (rule "=dfI"(2)[OF L_def])
5015     apply "cqt:2[lambda]"
5016    apply (rule "beta-C-meta"[THEN "→E"])
5017  by "cqt:2[lambda]"
5018
5019private AOT_lemma eqnotnec_123_Aux_ω: ‹[λz φ]x ≡ φ›
5020    by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5021
5022private AOT_lemma eqnotnec_123_Aux_θ: ‹φ ≡ ∀x([L]x ≡ [λz φ]x)›
5023proof(rule "≡I"; rule "→I"; (rule "∀I")?)
5024  fix x
5025  AOT_assume 1: ‹φ›
5026  AOT_have ‹[L]x ≡ (E!x → E!x)› using eqnotnec_123_Aux_ζ.
5027  also AOT_have ‹… ≡ φ›
5028    using "if-p-then-p" 1 "≡I" "→I" by simp
5029  also AOT_have ‹… ≡ [λz φ]x›
5030    using "Commutativity of ≡"[THEN "≡E"(1)] eqnotnec_123_Aux_ω by blast
5031  finally AOT_show ‹[L]x ≡ [λz φ]x›.
5032next
5033  fix x
5034  AOT_assume ‹∀x([L]x ≡ [λz φ]x)›
5035  AOT_hence ‹[L]x ≡ [λz φ]x› using "∀E" by blast
5036  also AOT_have ‹… ≡ φ› using eqnotnec_123_Aux_ω.
5037  finally AOT_have ‹φ ≡ [L]x› using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5038  also AOT_have ‹… ≡ E!x → E!x› using eqnotnec_123_Aux_ζ.
5039  finally AOT_show ‹φ› using "≡E" "if-p-then-p" by fast
5040qed
5041private lemmas eqnotnec_123_Aux_ξ =  eqnotnec_123_Aux_θ[THEN "oth-class-taut:4:b"[THEN "≡E"(1)],
5042                      THEN "conventions:3"[THEN "≡Df", THEN "≡E"(1), THEN "&E"(1)],
5043                      THEN "RM◇"]
5044private lemmas eqnotnec_123_Aux_ξ' = eqnotnec_123_Aux_θ[THEN "conventions:3"[THEN "≡Df", THEN "≡E"(1), THEN "&E"(1)], THEN "RM◇"]
5045
5046AOT_theorem "eqnotnec:1": ‹∃F∃G(∀x([F]x ≡ [G]x) & ◇¬∀x([F]x ≡ [G]x))›
5047proof-
5048  AOT_obtain p1 where ‹ContingentlyTrue(p1)› using "cont-tf-thm:1" "∃E"[rotated] by blast
5049  AOT_hence ‹p1 & ◇¬p1› using "cont-tf:1"[THEN "≡dfE"] by blast
5050  AOT_hence ‹∀x ([L]x ≡ [λz p1]x) & ◇¬∀x([L]x ≡ [λz p1]x)›
5051    apply - apply (rule "&I")
5052    using "&E" eqnotnec_123_Aux_θ[THEN "≡E"(1)] eqnotnec_123_Aux_ξ "→E" by fast+
5053  AOT_hence ‹∃G (∀x([L]x ≡ [G]x) & ◇¬∀x([L]x ≡ [G]x))›
5054    by (rule "∃I") "cqt:2[lambda]"
5055  AOT_thus ‹∃F∃G (∀x([F]x ≡ [G]x) & ◇¬∀x([F]x ≡ [G]x))›
5056    apply (rule "∃I")
5057    by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
5058qed
5059
5060AOT_theorem "eqnotnec:2": ‹∃F∃G(¬∀x([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5061proof-
5062  AOT_obtain p1 where ‹ContingentlyFalse(p1)› using "cont-tf-thm:2" "∃E"[rotated] by blast
5063  AOT_hence ‹¬p1 & ◇p1› using "cont-tf:2"[THEN "≡dfE"] by blast
5064  AOT_hence ‹¬∀x ([L]x ≡ [λz p1]x) & ◇∀x([L]x ≡ [λz p1]x)›
5065    apply - apply (rule "&I")
5066    using "&E" eqnotnec_123_Aux_θ[THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1)] eqnotnec_123_Aux_ξ' "→E" by fast+
5067  AOT_hence ‹∃G (¬∀x([L]x ≡ [G]x) & ◇∀x([L]x ≡ [G]x))›
5068    by (rule "∃I") "cqt:2[lambda]"
5069  AOT_thus ‹∃F∃G (¬∀x([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5070    apply (rule "∃I")
5071    by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
5072qed
5073
5074AOT_theorem "eqnotnec:3": ‹∃F∃G(𝒜¬∀x([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5075proof-
5076  AOT_have ‹¬𝒜q0›
5077    apply (rule "=dfI"(2)[OF q0_def])
5078     apply (fact "log-prop-prop:2")
5079    by (fact AOT)
5080  AOT_hence ‹𝒜¬q0›
5081    using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] by blast
5082  AOT_hence ‹𝒜¬∀x ([L]x ≡ [λz q0]x)›
5083    using eqnotnec_123_Aux_θ[THEN "oth-class-taut:4:b"[THEN "≡E"(1)],
5084            THEN "conventions:3"[THEN "≡Df", THEN "≡E"(1), THEN "&E"(1)],
5085            THEN "RA[2]", THEN "act-cond"[THEN "→E"], THEN "→E"] by blast
5086  moreover AOT_have ‹◇∀x ([L]x ≡ [λz q0]x)› using eqnotnec_123_Aux_ξ'[THEN "→E"] q0_prop[THEN "&E"(1)] by blast
5087  ultimately AOT_have ‹𝒜¬∀x ([L]x ≡ [λz q0]x) & ◇∀x ([L]x ≡ [λz q0]x)› using "&I" by blast
5088  AOT_hence ‹∃G (𝒜¬∀x([L]x ≡ [G]x) & ◇∀x([L]x ≡ [G]x))›
5089    by (rule "∃I") "cqt:2[lambda]"
5090  AOT_thus ‹∃F∃G (𝒜¬∀x([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5091    apply (rule "∃I")
5092    by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
5093qed
5094
5095end
5096
5097(* TODO[IMPORTANT]: proof of 219.4 ζ: appeal to (159.2) requires a theorem, but the result has local
5098   assumptions! *)
5099AOT_theorem "eqnotnec:4": ‹∀F∃G(∀x([F]x ≡ [G]x) & ◇¬∀x([F]x ≡ [G]x))›
5100proof(rule GEN)
5101  fix F
5102
5103  AOT_have Aux_A: ‹⊢□ ψ → ∀x([F]x ≡ [λz [F]z & ψ]x)› for ψ
5104  proof(rule "→I"; rule GEN)
5105    AOT_modally_strict {
5106    fix x
5107    AOT_assume 0: ‹ψ›
5108    AOT_have ‹[λz [F]z & ψ]x ≡ [F]x & ψ›
5109      by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5110    also AOT_have ‹... ≡ [F]x›
5111      apply (rule "≡I"; rule "→I")
5112      using  "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5113      using 0 "&I" by blast
5114    finally AOT_show ‹[F]x ≡ [λz [F]z & ψ]x›
5115      using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5116    }
5117  qed
5118
5119  AOT_have Aux_B: ‹⊢□ ψ → ∀x([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› for ψ
5120  proof (rule "→I"; rule GEN)
5121    AOT_modally_strict {
5122      fix x
5123      AOT_assume 0: ‹ψ›
5124      AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]x ≡ (([F]x & ψ) ∨ ¬ψ)›
5125        by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5126      also AOT_have ‹... ≡ [F]x›
5127        apply (rule "≡I"; rule "→I")
5128        using  "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5129        apply (rule "∨I"(1)) using 0 "&I" by blast
5130      finally AOT_show ‹[F]x ≡ [λz ([F]z & ψ) ∨ ¬ψ]x›
5131        using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5132    }
5133  qed
5134
5135  AOT_have Aux_C: ‹⊢□ ◇¬ψ → ◇¬∀z([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)› for ψ
5136  proof(rule "RM◇"; rule "→I"; rule "raa-cor:2")
5137  AOT_modally_strict {
5138      AOT_assume 0: ‹¬ψ›
5139      AOT_assume ‹∀z ([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)›
5140      AOT_hence ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5141      moreover AOT_have ‹[λz [F]z & ψ]z ≡ [F]z & ψ› for z
5142          by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5143      moreover AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]z ≡ (([F]z & ψ) ∨ ¬ψ)› for z
5144        by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5145      ultimately AOT_have ‹[F]z & ψ ≡ (([F]z & ψ) ∨ ¬ψ)› for z
5146        using "Commutativity of ≡"[THEN "≡E"(1)] "≡E"(5) by meson
5147      moreover AOT_have ‹(([F]z & ψ) ∨ ¬ψ)› for z using 0 "∨I" by blast
5148      ultimately AOT_have ‹ψ› using "≡E" "&E" by metis
5149      AOT_thus ‹ψ & ¬ψ› using 0 "&I" by blast
5150    }
5151  qed
5152
5153  AOT_have Aux_D: ‹□∀z ([F]z ≡ [λz [F]z & ψ]z) → (◇¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡ ◇¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x))› for ψ
5154  proof (rule "→I")
5155    AOT_assume A: ‹□∀z([F]z ≡ [λz [F]z & ψ]z)›
5156    AOT_show ‹◇¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡ ◇¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5157    proof(rule "≡I"; rule "KBasic:13"[THEN "→E"];
5158          rule "RN[prem]"[where Γ="{«∀z([F]z ≡ [λz [F]z & ψ]z)»}", simplified];
5159          (rule "useful-tautologies:5"[THEN "→E"]; rule "→I")?)
5160      AOT_modally_strict {
5161        AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5162        AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z using "∀E" by blast
5163        AOT_assume ‹∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5164        AOT_hence 2: ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5165        AOT_have ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "≡E" 1 2 by meson
5166        AOT_thus ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› by (rule GEN)
5167      }
5168    next
5169      AOT_modally_strict {
5170        AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5171        AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z using "∀E" by blast
5172        AOT_assume ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5173        AOT_hence 2: ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5174        AOT_have ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using 1 2 "≡E" by meson
5175        AOT_thus ‹ ∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› by (rule GEN)
5176      }
5177    qed(auto simp: A)
5178  qed
5179
5180  AOT_obtain p1 where p1_prop: ‹p1 & ◇¬p1› using "cont-tf-thm:1" "∃E"[rotated] "cont-tf:1"[THEN "≡dfE"] by blast
5181  {
5182    AOT_assume 1: ‹□∀x([F]x ≡ [λz [F]z & p1]x)›
5183    AOT_have 2: ‹∀x([F]x ≡ [λz [F]z & p1 ∨ ¬p1]x)›
5184      using Aux_B[THEN "→E", OF p1_prop[THEN "&E"(1)]].
5185    AOT_have ‹◇¬∀x([λz [F]z & p1]x ≡ [λz [F]z & p1 ∨ ¬p1]x)›
5186      using Aux_C[THEN "→E", OF p1_prop[THEN "&E"(2)]].
5187    AOT_hence 3: ‹◇¬∀x([F]x ≡ [λz [F]z & p1 ∨ ¬p1]x)›
5188      using Aux_D[THEN "→E", OF 1, THEN "≡E"(1)] by blast
5189    AOT_hence ‹∀x([F]x ≡ [λz [F]z & p1 ∨ ¬p1]x) & ◇¬∀x([F]x ≡ [λz [F]z & p1 ∨ ¬p1]x)› using 2 "&I" by blast
5190    AOT_hence ‹∃G (∀x ([F]x ≡ [G]x) & ◇¬∀x([F]x ≡ [G]x))›
5191      by (rule "∃I"(1)) "cqt:2[lambda]"
5192  }
5193  moreover {
5194    AOT_assume 2: ‹¬□∀x([F]x ≡ [λz [F]z & p1]x)›
5195    AOT_hence ‹◇¬∀x([F]x ≡ [λz [F]z & p1]x)›
5196      using "KBasic:11"[THEN "≡E"(1)] by blast
5197    AOT_hence ‹∀x ([F]x ≡ [λz [F]z & p1]x) & ◇¬∀x([F]x ≡ [λz [F]z & p1]x)›
5198      using Aux_A[THEN "→E", OF p1_prop[THEN "&E"(1)]] "&I" by blast
5199    AOT_hence ‹∃G (∀x ([F]x ≡ [G]x) & ◇¬∀x([F]x ≡ [G]x))›
5200      by (rule "∃I"(1)) "cqt:2[lambda]"
5201  }
5202  ultimately AOT_show ‹∃G (∀x ([F]x ≡ [G]x) & ◇¬∀x([F]x ≡ [G]x))›
5203    using "∨E"(1)[OF "exc-mid"] "→I" by blast
5204qed
5205
5206AOT_theorem "eqnotnec:5": ‹∀F∃G(¬∀x([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5207proof(rule GEN)
5208  fix F
5209
5210  AOT_have Aux_A: ‹⊢□ ◇ψ → ◇∀x([F]x ≡ [λz [F]z & ψ]x)› for ψ
5211  proof(rule "RM◇"; rule "→I"; rule GEN)
5212    AOT_modally_strict {
5213    fix x
5214    AOT_assume 0: ‹ψ›
5215    AOT_have ‹[λz [F]z & ψ]x ≡ [F]x & ψ›
5216      by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5217    also AOT_have ‹... ≡ [F]x›
5218      apply (rule "≡I"; rule "→I")
5219      using  "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5220      using 0 "&I" by blast
5221    finally AOT_show ‹[F]x ≡ [λz [F]z & ψ]x›
5222      using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5223    }
5224  qed
5225
5226  AOT_have Aux_B: ‹⊢□ ◇ψ → ◇∀x([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› for ψ
5227  proof (rule "RM◇"; rule "→I"; rule GEN)
5228    AOT_modally_strict {
5229      fix x
5230      AOT_assume 0: ‹ψ›
5231      AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]x ≡ (([F]x & ψ) ∨ ¬ψ)›
5232        by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5233      also AOT_have ‹... ≡ [F]x›
5234        apply (rule "≡I"; rule "→I")
5235        using  "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5236        apply (rule "∨I"(1)) using 0 "&I" by blast
5237      finally AOT_show ‹[F]x ≡ [λz ([F]z & ψ) ∨ ¬ψ]x›
5238        using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5239    }
5240  qed
5241
5242  AOT_have Aux_C: ‹⊢□ ¬ψ → ¬∀z([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)› for ψ
5243  proof(rule "→I"; rule "raa-cor:2")
5244  AOT_modally_strict {
5245      AOT_assume 0: ‹¬ψ›
5246      AOT_assume ‹∀z ([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)›
5247      AOT_hence ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5248      moreover AOT_have ‹[λz [F]z & ψ]z ≡ [F]z & ψ› for z
5249          by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5250      moreover AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]z ≡ (([F]z & ψ) ∨ ¬ψ)› for z
5251        by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5252      ultimately AOT_have ‹[F]z & ψ ≡ (([F]z & ψ) ∨ ¬ψ)› for z
5253        using "Commutativity of ≡"[THEN "≡E"(1)] "≡E"(5) by meson
5254      moreover AOT_have ‹(([F]z & ψ) ∨ ¬ψ)› for z using 0 "∨I" by blast
5255      ultimately AOT_have ‹ψ› using "≡E" "&E" by metis
5256      AOT_thus ‹ψ & ¬ψ› using 0 "&I" by blast
5257    }
5258  qed
5259
5260  AOT_have Aux_D: ‹∀z ([F]z ≡ [λz [F]z & ψ]z) → (¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡ ¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x))› for ψ
5261  proof (rule "→I"; rule "≡I"; (rule "useful-tautologies:5"[THEN "→E"]; rule "→I")?)
5262    AOT_modally_strict {
5263      AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5264      AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z using "∀E" by blast
5265      AOT_assume ‹∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5266      AOT_hence 2: ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5267      AOT_have ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "≡E" 1 2 by meson
5268      AOT_thus ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› by (rule GEN)
5269    }
5270  next
5271    AOT_modally_strict {
5272      AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5273      AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z using "∀E" by blast
5274      AOT_assume ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5275      AOT_hence 2: ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5276      AOT_have ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using 1 2 "≡E" by meson
5277      AOT_thus ‹ ∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› by (rule GEN)
5278    }
5279  qed
5280
5281  AOT_obtain p1 where p1_prop: ‹¬p1 & ◇p1› using "cont-tf-thm:2" "∃E"[rotated] "cont-tf:2"[THEN "≡dfE"] by blast
5282  {
5283    AOT_assume 1: ‹∀x([F]x ≡ [λz [F]z & p1]x)›
5284    AOT_have 2: ‹◇∀x([F]x ≡ [λz [F]z & p1 ∨ ¬p1]x)›
5285      using Aux_B[THEN "→E", OF p1_prop[THEN "&E"(2)]].
5286    AOT_have ‹¬∀x([λz [F]z & p1]x ≡ [λz [F]z & p1 ∨ ¬p1]x)›
5287      using Aux_C[THEN "→E", OF p1_prop[THEN "&E"(1)]].
5288    AOT_hence 3: ‹¬∀x([F]x ≡ [λz [F]z & p1 ∨ ¬p1]x)›
5289      using Aux_D[THEN "→E", OF 1, THEN "≡E"(1)] by blast
5290    AOT_hence ‹¬∀x([F]x ≡ [λz [F]z & p1 ∨ ¬p1]x) & ◇∀x([F]x ≡ [λz [F]z & p1 ∨ ¬p1]x)› using 2 "&I" by blast
5291    AOT_hence ‹∃G (¬∀x ([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5292      by (rule "∃I"(1)) "cqt:2[lambda]"
5293  }
5294  moreover {
5295    AOT_assume 2: ‹¬∀x([F]x ≡ [λz [F]z & p1]x)›
5296    AOT_hence ‹¬∀x([F]x ≡ [λz [F]z & p1]x)›
5297      using "KBasic:11"[THEN "≡E"(1)] by blast
5298    AOT_hence ‹¬∀x ([F]x ≡ [λz [F]z & p1]x) & ◇∀x([F]x ≡ [λz [F]z & p1]x)›
5299      using Aux_A[THEN "→E", OF p1_prop[THEN "&E"(2)]] "&I" by blast
5300    AOT_hence ‹∃G (¬∀x ([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5301      by (rule "∃I"(1)) "cqt:2[lambda]"
5302  }
5303  ultimately AOT_show ‹∃G (¬∀x ([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5304    using "∨E"(1)[OF "exc-mid"] "→I" by blast
5305qed
5306
5307AOT_theorem "eqnotnec:6": ‹∀F∃G(𝒜¬∀x([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5308proof(rule GEN)
5309  fix F
5310
5311  AOT_have Aux_A: ‹⊢□ ◇ψ → ◇∀x([F]x ≡ [λz [F]z & ψ]x)› for ψ
5312  proof(rule "RM◇"; rule "→I"; rule GEN)
5313    AOT_modally_strict {
5314    fix x
5315    AOT_assume 0: ‹ψ›
5316    AOT_have ‹[λz [F]z & ψ]x ≡ [F]x & ψ›
5317      by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5318    also AOT_have ‹... ≡ [F]x›
5319      apply (rule "≡I"; rule "→I")
5320      using  "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5321      using 0 "&I" by blast
5322    finally AOT_show ‹[F]x ≡ [λz [F]z & ψ]x›
5323      using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5324    }
5325  qed
5326
5327  AOT_have Aux_B: ‹⊢□ ◇ψ → ◇∀x([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› for ψ
5328  proof (rule "RM◇"; rule "→I"; rule GEN)
5329    AOT_modally_strict {
5330      fix x
5331      AOT_assume 0: ‹ψ›
5332      AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]x ≡ (([F]x & ψ) ∨ ¬ψ)›
5333        by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5334      also AOT_have ‹... ≡ [F]x›
5335        apply (rule "≡I"; rule "→I")
5336        using  "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5337        apply (rule "∨I"(1)) using 0 "&I" by blast
5338      finally AOT_show ‹[F]x ≡ [λz ([F]z & ψ) ∨ ¬ψ]x›
5339        using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5340    }
5341  qed
5342
5343  AOT_have Aux_C: ‹⊢□ 𝒜¬ψ → 𝒜¬∀z([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)› for ψ
5344  proof(rule "act-cond"[THEN "→E"]; rule "RA[2]"; rule "→I"; rule "raa-cor:2")
5345  AOT_modally_strict {
5346      AOT_assume 0: ‹¬ψ›
5347      AOT_assume ‹∀z ([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)›
5348      AOT_hence ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5349      moreover AOT_have ‹[λz [F]z & ψ]z ≡ [F]z & ψ› for z
5350          by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5351      moreover AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]z ≡ (([F]z & ψ) ∨ ¬ψ)› for z
5352        by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5353      ultimately AOT_have ‹[F]z & ψ ≡ (([F]z & ψ) ∨ ¬ψ)› for z
5354        using "Commutativity of ≡"[THEN "≡E"(1)] "≡E"(5) by meson
5355      moreover AOT_have ‹(([F]z & ψ) ∨ ¬ψ)› for z using 0 "∨I" by blast
5356      ultimately AOT_have ‹ψ› using "≡E" "&E" by metis
5357      AOT_thus ‹ψ & ¬ψ› using 0 "&I" by blast
5358    }
5359  qed
5360
5361  AOT_have Aux_D: ‹𝒜∀z ([F]z ≡ [λz [F]z & ψ]z) → (𝒜¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡ 𝒜¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x))› for ψ
5362  proof (rule "→I"; rule "Act-Basic:5"[THEN "≡E"(1)])
5363    AOT_assume ‹𝒜∀z ([F]z ≡ [λz [F]z & ψ]z)›
5364    AOT_thus ‹𝒜(¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡ ¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x))›
5365    proof (rule "RA[3]"[where Γ="{«∀z ([F]z ≡ [λz [F]z & ψ]z)»}", simplified, rotated])
5366      AOT_modally_strict {
5367        AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5368        AOT_thus ‹¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡ ¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5369          apply -
5370        proof(rule "≡I"; (rule "useful-tautologies:5"[THEN "→E"]; rule "→I")?)
5371        AOT_modally_strict {
5372          AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5373          AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z using "∀E" by blast
5374          AOT_assume ‹∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5375          AOT_hence 2: ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5376          AOT_have ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "≡E" 1 2 by meson
5377          AOT_thus ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› by (rule GEN)
5378        }
5379      next
5380        AOT_modally_strict {
5381          AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5382          AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z using "∀E" by blast
5383          AOT_assume ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5384          AOT_hence 2: ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5385          AOT_have ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using 1 2 "≡E" by meson
5386          AOT_thus ‹ ∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› by (rule GEN)
5387        }
5388      qed
5389      }
5390    qed
5391  qed
5392
5393  AOT_have ‹¬𝒜q0›
5394    apply (rule "=dfI"(2)[OF q0_def])
5395     apply (fact "log-prop-prop:2")
5396    by (fact AOT)
5397  AOT_hence q0_prop_1: ‹𝒜¬q0›
5398    using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] by blast
5399  {
5400    AOT_assume 1: ‹𝒜∀x([F]x ≡ [λz [F]z & q0]x)›
5401    AOT_have 2: ‹◇∀x([F]x ≡ [λz [F]z & q0 ∨ ¬q0]x)›
5402      using Aux_B[THEN "→E", OF q0_prop[THEN "&E"(1)]].
5403    AOT_have ‹𝒜¬∀x([λz [F]z & q0]x ≡ [λz [F]z & q0 ∨ ¬q0]x)›
5404      using Aux_C[THEN "→E", OF q0_prop_1].
5405    AOT_hence 3: ‹𝒜¬∀x([F]x ≡ [λz [F]z & q0 ∨ ¬q0]x)›
5406      using Aux_D[THEN "→E", OF 1, THEN "≡E"(1)] by blast
5407    AOT_hence ‹𝒜¬∀x([F]x ≡ [λz [F]z & q0 ∨ ¬q0]x) & ◇∀x([F]x ≡ [λz [F]z & q0 ∨ ¬q0]x)› using 2 "&I" by blast
5408    AOT_hence ‹∃G (𝒜¬∀x ([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5409      by (rule "∃I"(1)) "cqt:2[lambda]"
5410  }
5411  moreover {
5412    AOT_assume 2: ‹¬𝒜∀x([F]x ≡ [λz [F]z & q0]x)›
5413    AOT_hence ‹𝒜¬∀x([F]x ≡ [λz [F]z & q0]x)›
5414      using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] by blast
5415    AOT_hence ‹𝒜¬∀x ([F]x ≡ [λz [F]z & q0]x) & ◇∀x([F]x ≡ [λz [F]z & q0]x)›
5416      using Aux_A[THEN "→E", OF q0_prop[THEN "&E"(1)]] "&I" by blast
5417    AOT_hence ‹∃G (𝒜¬∀x ([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5418      by (rule "∃I"(1)) "cqt:2[lambda]"
5419  }
5420  ultimately AOT_show ‹∃G (𝒜¬∀x ([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5421    using "∨E"(1)[OF "exc-mid"] "→I" by blast
5422qed
5423
5424AOT_theorem "oa-contingent:1": ‹O! ≠ A!›
5425proof(rule "≡dfI"[OF "=-infix"]; rule "raa-cor:2")
5426  fix x
5427  AOT_assume 1: ‹O! = A!›
5428  AOT_hence ‹[λx ◇E!x] = A!›
5429    by (rule "=dfE"(2)[OF AOT_ordinary, rotated]) "cqt:2[lambda]"
5430  AOT_hence ‹[λx ◇E!x] = [λx ¬◇E!x]›
5431    by (rule "=dfE"(2)[OF AOT_abstract, rotated]) "cqt:2[lambda]"
5432  moreover AOT_have ‹[λx ◇E!x]x ≡ ◇E!x›
5433    by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5434  ultimately AOT_have ‹[λx ¬◇E!x]x ≡ ◇E!x›
5435    using "rule=E" by fast
5436  moreover AOT_have ‹[λx ¬◇E!x]x ≡ ¬◇E!x›
5437    by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5438  ultimately AOT_have ‹◇E!x ≡ ¬◇E!x› using "≡E"(6) "Commutativity of ≡"[THEN "≡E"(1)] by blast
5439  AOT_thus "(◇E!x ≡ ¬◇E!x) & ¬(◇E!x ≡ ¬◇E!x)" using "oth-class-taut:3:c" "&I" by blast
5440qed
5441
5442AOT_theorem "oa-contingent:2": ‹O!x ≡ ¬A!x›
5443proof -
5444  AOT_have ‹O!x ≡ [λx ◇E!x]x›
5445    apply (rule "≡I"; rule "→I")
5446     apply (rule "=dfE"(2)[OF AOT_ordinary])
5447      apply "cqt:2[lambda]"
5448     apply argo
5449    apply (rule  "=dfI"(2)[OF AOT_ordinary])
5450     apply "cqt:2[lambda]"
5451    by argo
5452  also AOT_have ‹… ≡ ◇E!x›
5453    by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5454  also AOT_have ‹… ≡ ¬¬◇E!x›
5455    using "oth-class-taut:3:b".
5456  also AOT_have ‹… ≡ ¬[λx ¬◇E!x]x›
5457    by (rule "beta-C-meta"[THEN "→E", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], symmetric]) "cqt:2[lambda]"
5458  also AOT_have ‹… ≡ ¬A!x›
5459    apply (rule "≡I"; rule "→I")
5460     apply (rule "=dfI"(2)[OF AOT_abstract])
5461      apply "cqt:2[lambda]"
5462     apply argo
5463    apply (rule "=dfE"(2)[OF AOT_abstract])
5464     apply "cqt:2[lambda]"
5465    by argo
5466  finally show ?thesis.
5467qed
5468
5469AOT_theorem "oa-contingent:3": ‹A!x ≡ ¬O!x›
5470  by (AOT_subst "«A!x»" "«¬¬A!x»")
5471     (auto simp add: "oth-class-taut:3:b" "oa-contingent:2"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)], symmetric])
5472
5473AOT_theorem "oa-contingent:4": ‹Contingent(O!)›
5474proof (rule "thm-cont-prop:2"[unvarify F, OF "oa-exist:1", THEN "≡E"(2)]; rule "&I")
5475  AOT_have ‹◇∃x E!x› using "thm-cont-e:3" .
5476  AOT_hence ‹∃x ◇E!x› using "BF◇"[THEN "→E"] by blast
5477  then AOT_obtain a where ‹◇E!a› using "∃E"[rotated] by blast
5478  AOT_hence ‹[λx ◇E!x]a›
5479    by (rule "beta-C-meta"[THEN "→E", THEN "≡E"(2), rotated]) "cqt:2[lambda]"
5480  AOT_hence ‹O!a›
5481    by (rule "=dfI"(2)[OF AOT_ordinary, rotated]) "cqt:2[lambda]"
5482  AOT_hence ‹∃x O!x› using "∃I" by blast
5483  AOT_thus ‹◇∃x O!x› using "T◇"[THEN "→E"] by blast
5484next
5485  AOT_obtain a where ‹A!a›
5486    using "A-objects"[axiom_inst] "∃E"[rotated] "&E" by blast
5487  AOT_hence ‹¬O!a› using "oa-contingent:3"[THEN "≡E"(1)] by blast
5488  AOT_hence ‹∃x ¬O!x› using "∃I" by fast
5489  AOT_thus ‹◇∃x ¬O!x› using "T◇"[THEN "→E"] by blast
5490qed
5491
5492AOT_theorem "oa-contingent:5": ‹Contingent(A!)›
5493proof (rule "thm-cont-prop:2"[unvarify F, OF "oa-exist:2", THEN "≡E"(2)]; rule "&I")
5494  AOT_obtain a where ‹A!a›
5495    using "A-objects"[axiom_inst] "∃E"[rotated] "&E" by blast
5496  AOT_hence ‹∃x A!x› using "∃I" by fast
5497  AOT_thus ‹◇∃x A!x› using "T◇"[THEN "→E"] by blast
5498next
5499  AOT_have ‹◇∃x E!x› using "thm-cont-e:3" .
5500  AOT_hence ‹∃x ◇E!x› using "BF◇"[THEN "→E"] by blast
5501  then AOT_obtain a where ‹◇E!a› using "∃E"[rotated] by blast
5502  AOT_hence ‹[λx ◇E!x]a›
5503    by (rule "beta-C-meta"[THEN "→E", THEN "≡E"(2), rotated]) "cqt:2[lambda]"
5504  AOT_hence ‹O!a›
5505    by (rule "=dfI"(2)[OF AOT_ordinary, rotated]) "cqt:2[lambda]"
5506  AOT_hence ‹¬A!a› using "oa-contingent:2"[THEN "≡E"(1)] by blast
5507  AOT_hence ‹∃x ¬A!x› using "∃I" by fast
5508  AOT_thus ‹◇∃x ¬A!x› using "T◇"[THEN "→E"] by blast
5509qed
5510
5511AOT_theorem "oa-contingent:7": ‹O!-x ≡ ¬A!-x›
5512proof -
5513  AOT_have ‹O!x ≡ ¬A!x›
5514    using "oa-contingent:2" by blast
5515  also AOT_have ‹… ≡ A!-x›
5516    using "thm-relation-negation:1"[symmetric, unvarify F, OF "oa-exist:2"].
5517  finally AOT_have 1: ‹O!x ≡ A!-x›.
5518
5519  AOT_have ‹A!x ≡ ¬O!x›
5520    using "oa-contingent:3" by blast
5521  also AOT_have ‹… ≡ O!-x›
5522    using "thm-relation-negation:1"[symmetric, unvarify F, OF "oa-exist:1"].
5523  finally AOT_have 2: ‹A!x ≡ O!-x›.
5524
5525  AOT_show ‹O!-x ≡ ¬A!-x›
5526    using 1[THEN "oth-class-taut:4:b"[THEN "≡E"(1)]] "oa-contingent:3"[of _ x] 2[symmetric]
5527          "≡E"(5) by blast
5528qed
5529
5530AOT_theorem "oa-contingent:6": ‹O!- ≠ A!-›
5531proof (rule "=-infix"[THEN "≡dfI"]; rule "raa-cor:2")
5532  AOT_assume 1: ‹O!- = A!-›
5533  fix x
5534  AOT_have ‹A!-x ≡ O!-x›
5535    apply (rule "rule=E"[rotated, OF 1]) by (fact "oth-class-taut:3:a")
5536  AOT_hence ‹A!-x ≡ ¬A!-x›
5537    using "oa-contingent:7" "≡E" by fast
5538  AOT_thus ‹(A!-x ≡ ¬A!-x) & ¬(A!-x ≡ ¬A!-x)› using "oth-class-taut:3:c" "&I" by blast
5539qed
5540
5541AOT_theorem "oa-contingent:8": ‹Contingent(O!-)›
5542  using "thm-cont-prop:3"[unvarify F, OF "oa-exist:1", THEN "≡E"(1), OF "oa-contingent:4"].
5543
5544AOT_theorem "oa-contingent:9": ‹Contingent(A!-)›
5545  using "thm-cont-prop:3"[unvarify F, OF "oa-exist:2", THEN "≡E"(1), OF "oa-contingent:5"].
5546
5547AOT_define WeaklyContingent :: ‹Π ⇒ φ› ("WeaklyContingent'(_')")
5548  "df-cont-nec": "WeaklyContingent([F]) ≡df Contingent([F]) & ∀x (◇[F]x → □[F]x)"
5549
5550AOT_theorem "cont-nec-fact1:1": ‹WeaklyContingent([F]) ≡ WeaklyContingent([F]-)›
5551proof -
5552  AOT_have ‹WeaklyContingent([F]) ≡ Contingent([F]) & ∀x (◇[F]x → □[F]x)›
5553    using "df-cont-nec"[THEN "≡Df"] by blast
5554  also AOT_have ‹... ≡ Contingent([F]-) & ∀x (◇[F]x → □[F]x)›
5555    apply (rule "oth-class-taut:8:f"[THEN "≡E"(2)]; rule "→I")
5556    using "thm-cont-prop:3".
5557  also AOT_have ‹… ≡ Contingent([F]-) & ∀x (◇[F]-x → □[F]-x)›
5558  proof (rule "oth-class-taut:8:e"[THEN "≡E"(2)]; rule "→I"; rule "≡I"; rule "→I"; rule GEN; rule "→I")
5559    fix x
5560    AOT_assume 0: ‹∀x (◇[F]x → □[F]x)›
5561    AOT_assume 1: ‹◇[F]-x›
5562    AOT_have ‹◇¬[F]x›
5563      by (AOT_subst_rev "«[F]-x»" "«¬[F]x»")
5564         (auto simp add: "thm-relation-negation:1" 1)
5565    AOT_hence 2: ‹¬□[F]x›
5566      using "KBasic:11"[THEN "≡E"(2)] by blast
5567    AOT_show ‹□[F]-x›
5568    proof (rule "raa-cor:1")
5569      AOT_assume 3: ‹¬□[F]-x›
5570      AOT_have ‹¬□¬[F]x›
5571        by (AOT_subst_rev "«[F]-x»" "«¬[F]x»")
5572           (auto simp add: "thm-relation-negation:1" 3)
5573      AOT_hence ‹◇[F]x›
5574        using "conventions:5"[THEN "≡dfI"] by simp
5575      AOT_hence ‹□[F]x› using 0 "∀E" "→E" by fast
5576      AOT_thus ‹□[F]x & ¬□[F]x› using "&I" 2 by blast
5577    qed
5578  next
5579    fix x
5580    AOT_assume 0: ‹∀x (◇[F]-x → □[F]-x)›
5581    AOT_assume 1: ‹◇[F]x›
5582    AOT_have ‹◇¬[F]-x›
5583      by (AOT_subst "«¬[F]-x»" "«[F]x»")
5584         (auto simp: "thm-relation-negation:2" 1)
5585    AOT_hence 2: ‹¬□[F]-x›
5586      using "KBasic:11"[THEN "≡E"(2)] by blast
5587    AOT_show ‹□[F]x›
5588    proof (rule "raa-cor:1")
5589      AOT_assume 3: ‹¬□[F]x›
5590      AOT_have ‹¬□¬[F]-x›
5591        by (AOT_subst "«¬[F]-x»" "«[F]x»")
5592           (auto simp add: "thm-relation-negation:2" 3)
5593      AOT_hence ‹◇[F]-x›
5594        using "conventions:5"[THEN "≡dfI"] by simp
5595      AOT_hence ‹□[F]-x› using 0 "∀E" "→E" by fast
5596      AOT_thus ‹□[F]-x & ¬□[F]-x› using "&I" 2 by blast
5597    qed
5598  qed
5599  also AOT_have ‹… ≡ WeaklyContingent([F]-)›
5600    using "df-cont-nec"[THEN "≡Df", symmetric] by blast
5601  finally show ?thesis.
5602qed
5603
5604AOT_theorem "cont-nec-fact1:2": ‹(WeaklyContingent([F]) & ¬WeaklyContingent([G])) → F ≠ G›
5605proof (rule "→I"; rule "=-infix"[THEN "≡dfI"]; rule "raa-cor:2")
5606  AOT_assume 1: ‹WeaklyContingent([F]) & ¬WeaklyContingent([G])›
5607  AOT_hence ‹WeaklyContingent([F])› using "&E" by blast
5608  moreover AOT_assume ‹F = G›
5609  ultimately AOT_have ‹WeaklyContingent([G])›
5610    using "rule=E" by blast
5611  AOT_thus ‹WeaklyContingent([G]) & ¬WeaklyContingent([G])›
5612    using 1 "&I" "&E" by blast
5613qed
5614
5615AOT_theorem "cont-nec-fact2:1": ‹WeaklyContingent(O!)›
5616proof (rule "df-cont-nec"[THEN "≡dfI"]; rule "&I")
5617  AOT_show ‹Contingent(O!)›
5618    using "oa-contingent:4".
5619next
5620  AOT_show ‹∀x (◇[O!]x → □[O!]x)›
5621    apply (rule GEN; rule "→I")
5622    using "oa-facts:5"[THEN "≡E"(1)] by blast
5623qed
5624
5625
5626AOT_theorem "cont-nec-fact2:2": ‹WeaklyContingent(A!)›
5627proof (rule "df-cont-nec"[THEN "≡dfI"]; rule "&I")
5628  AOT_show ‹Contingent(A!)›
5629    using "oa-contingent:5".
5630next
5631  AOT_show ‹∀x (◇[A!]x → □[A!]x)›
5632    apply (rule GEN; rule "→I")
5633    using "oa-facts:6"[THEN "≡E"(1)] by blast
5634qed
5635
5636AOT_theorem "cont-nec-fact2:3": ‹¬WeaklyContingent(E!)›
5637proof (rule "df-cont-nec"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)];
5638       rule DeMorgan(1)[THEN "≡E"(2)]; rule "∨I"(2); rule "raa-cor:2")
5639  AOT_have ‹◇∃x (E!x & ¬𝒜E!x)› using "qml:4"[axiom_inst].
5640  AOT_hence ‹∃x ◇(E!x & ¬𝒜E!x)› using "BF◇"[THEN "→E"] by blast
5641  then AOT_obtain a where ‹◇(E!a & ¬𝒜E!a)› using "∃E"[rotated] by blast
5642  AOT_hence 1: ‹◇E!a & ◇¬𝒜E!a› using "KBasic2:3"[THEN "→E"] by simp
5643  moreover AOT_assume ‹∀x (◇[E!]x → □[E!]x)›
5644  ultimately AOT_have ‹□E!a› using "&E" "∀E" "→E" by fast
5645  AOT_hence ‹𝒜E!a› using "nec-imp-act"[THEN "→E"] by blast
5646  AOT_hence ‹□𝒜E!a› using "qml-act:1"[axiom_inst, THEN "→E"] by blast
5647  moreover AOT_have ‹¬□𝒜E!a› using "KBasic:11"[THEN "≡E"(2)] 1[THEN "&E"(2)] by meson
5648  ultimately AOT_have ‹□𝒜E!a & ¬□𝒜E!a› using "&I" by blast
5649  AOT_thus ‹p & ¬p› for p using "raa-cor:1" by blast
5650qed
5651
5652AOT_theorem "cont-nec-fact2:4": ‹¬WeaklyContingent(L)›
5653  apply (rule "df-cont-nec"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)];
5654       rule DeMorgan(1)[THEN "≡E"(2)]; rule "∨I"(1))
5655  apply (rule "contingent-properties:4"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
5656  apply (rule DeMorgan(1)[THEN "≡E"(2)]; rule "∨I"(2); rule "useful-tautologies:2"[THEN "→E"])
5657  using "thm-noncont-e-e:3"[THEN "contingent-properties:3"[THEN "≡dfE"]].
5658
5659(* TODO: cleanup *)
5660AOT_theorem "cont-nec-fact2:5": ‹O! ≠ E! & O! ≠ E!- & O! ≠ L & O! ≠ L-›
5661proof -
5662  AOT_have 1: ‹L↓›
5663    by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
5664  {
5665    fix φ and Π and Π'
5666    AOT_have A: ‹¬(φ{Π'} ≡ φ{Π})› if  ‹φ{Π}› and ‹¬φ{Π'}›
5667    proof (rule "raa-cor:2")
5668      AOT_assume ‹φ{Π'} ≡ φ{Π}›
5669      AOT_hence ‹φ{Π'}› using that(1) "≡E" by blast
5670      AOT_thus ‹φ{Π'} & ¬φ{Π'}› using that(2) "&I" by blast
5671    qed
5672    AOT_have ‹Π' ≠ Π› if ‹Π↓› and ‹Π'↓› and ‹φ{Π}› and ‹¬φ{Π'}›
5673      using "pos-not-equiv-ne:4"[unvarify F G, THEN "→E", OF that(1,2), OF A[OF that(3, 4)]].
5674  } note 0 = this
5675  show ?thesis
5676    apply(safe intro!: "&I"; rule 0)
5677    using "cqt:2[concrete]"[axiom_inst] apply blast
5678    using "oa-exist:1" apply blast
5679    using "cont-nec-fact2:3" apply fast
5680    apply (rule "useful-tautologies:2"[THEN "→E"])
5681    using "cont-nec-fact2:1" apply fast
5682    using "rel-neg-T:3" apply fast
5683    using "oa-exist:1" apply blast
5684    using "cont-nec-fact1:1"[unvarify F, THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated, OF "cont-nec-fact2:3", OF "cqt:2[concrete]"[axiom_inst]] apply fast
5685    apply (rule "useful-tautologies:2"[THEN "→E"])
5686    using "cont-nec-fact2:1" apply blast
5687    apply (rule "=dfI"(2)[OF L_def]; "cqt:2[lambda]")
5688    using "oa-exist:1" apply fast
5689    using "cont-nec-fact2:4" apply fast
5690    apply (rule "useful-tautologies:2"[THEN "→E"])
5691    using "cont-nec-fact2:1" apply fast
5692    using "rel-neg-T:3" apply fast
5693    using "oa-exist:1" apply fast
5694    apply (rule "cont-nec-fact1:1"[unvarify F, THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated, OF "cont-nec-fact2:4"])
5695    apply (rule "=dfI"(2)[OF L_def]; "cqt:2[lambda]")
5696    apply (rule "useful-tautologies:2"[THEN "→E"])
5697    using "cont-nec-fact2:1" by blast
5698qed
5699
5700(* TODO: cleanup together with above *)
5701AOT_theorem "cont-nec-fact2:6": ‹A! ≠ E! & A! ≠ E!- & A! ≠ L & A! ≠ L-›
5702proof -
5703  AOT_have 1: ‹L↓›
5704    by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
5705  {
5706    fix φ and Π and Π'
5707    AOT_have A: ‹¬(φ{Π'} ≡ φ{Π})› if  ‹φ{Π}› and ‹¬φ{Π'}›
5708    proof (rule "raa-cor:2")
5709      AOT_assume ‹φ{Π'} ≡ φ{Π}›
5710      AOT_hence ‹φ{Π'}› using that(1) "≡E" by blast
5711      AOT_thus ‹φ{Π'} & ¬φ{Π'}› using that(2) "&I" by blast
5712    qed
5713    AOT_have ‹Π' ≠ Π› if ‹Π↓› and ‹Π'↓› and ‹φ{Π}› and ‹¬φ{Π'}›
5714      using "pos-not-equiv-ne:4"[unvarify F G, THEN "→E", OF that(1,2), OF A[OF that(3, 4)]].
5715  } note 0 = this
5716  show ?thesis
5717    apply(safe intro!: "&I"; rule 0)
5718    using "cqt:2[concrete]"[axiom_inst] apply blast
5719    using "oa-exist:2" apply blast
5720    using "cont-nec-fact2:3" apply fast
5721    apply (rule "useful-tautologies:2"[THEN "→E"])
5722    using "cont-nec-fact2:2" apply fast
5723    using "rel-neg-T:3" apply fast
5724    using "oa-exist:2" apply blast
5725    using "cont-nec-fact1:1"[unvarify F, THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated, OF "cont-nec-fact2:3", OF "cqt:2[concrete]"[axiom_inst]] apply fast
5726    apply (rule "useful-tautologies:2"[THEN "→E"])
5727    using "cont-nec-fact2:2" apply blast
5728    apply (rule "=dfI"(2)[OF L_def]; "cqt:2[lambda]")
5729    using "oa-exist:2" apply fast
5730    using "cont-nec-fact2:4" apply fast
5731    apply (rule "useful-tautologies:2"[THEN "→E"])
5732    using "cont-nec-fact2:2" apply fast
5733    using "rel-neg-T:3" apply fast
5734    using "oa-exist:2" apply fast
5735    apply (rule "cont-nec-fact1:1"[unvarify F, THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated, OF "cont-nec-fact2:4"])
5736    apply (rule "=dfI"(2)[OF L_def]; "cqt:2[lambda]")
5737    apply (rule "useful-tautologies:2"[THEN "→E"])
5738    using "cont-nec-fact2:2" by blast
5739qed
5740
5741AOT_define necessary_or_contingently_false :: ‹φ ⇒ φ› ("Δ_" [49] 54)
5742  ‹Δp ≡df □p ∨ (¬𝒜p & ◇p)›
5743
5744AOT_theorem sixteen:
5745 shows ‹∃F1∃F2∃F3∃F4∃F5∃F6∃F7∃F8∃F9∃F10∃F11∃F12∃F13∃F14∃F15∃F16 (
5746«F1::<κ>» ≠ F2 & F1 ≠ F3 & F1 ≠ F4 & F1 ≠ F5 & F1 ≠ F6 & F1 ≠ F7 & F1 ≠ F8 & F1 ≠ F9 & F1 ≠ F10 & F1 ≠ F11 & F1 ≠ F12 & F1 ≠ F13 & F1 ≠ F14 & F1 ≠ F15 & F1 ≠ F16 &
5747F2 ≠ F3 & F2 ≠ F4 & F2 ≠ F5 & F2 ≠ F6 & F2 ≠ F7 & F2 ≠ F8 & F2 ≠ F9 & F2 ≠ F10 & F2 ≠ F11 & F2 ≠ F12 & F2 ≠ F13 & F2 ≠ F14 & F2 ≠ F15 & F2 ≠ F16 &
5748F3 ≠ F4 & F3 ≠ F5 & F3 ≠ F6 & F3 ≠ F7 & F3 ≠ F8 & F3 ≠ F9 & F3 ≠ F10 & F3 ≠ F11 & F3 ≠ F12 & F3 ≠ F13 & F3 ≠ F14 & F3 ≠ F15 & F3 ≠ F16 &
5749F4 ≠ F5 & F4 ≠ F6 & F4 ≠ F7 & F4 ≠ F8 & F4 ≠ F9 & F4 ≠ F10 & F4 ≠ F11 & F4 ≠ F12 & F4 ≠ F13 & F4 ≠ F14 & F4 ≠ F15 & F4 ≠ F16 &
5750F5 ≠ F6 & F5 ≠ F7 & F5 ≠ F8 & F5 ≠ F9 & F5 ≠ F10 & F5 ≠ F11 & F5 ≠ F12 & F5 ≠ F13 & F5 ≠ F14 & F5 ≠ F15 & F5 ≠ F16 &
5751F6 ≠ F7 & F6 ≠ F8 & F6 ≠ F9 & F6 ≠ F10 & F6 ≠ F11 & F6 ≠ F12 & F6 ≠ F13 & F6 ≠ F14 & F6 ≠ F15 & F6 ≠ F16 &
5752F7 ≠ F8 & F7 ≠ F9 & F7 ≠ F10 & F7 ≠ F11 & F7 ≠ F12 & F7 ≠ F13 & F7 ≠ F14 & F7 ≠ F15 & F7 ≠ F16 &
5753F8 ≠ F9 & F8 ≠ F10 & F8 ≠ F11 & F8 ≠ F12 & F8 ≠ F13 & F8 ≠ F14 & F8 ≠ F15 & F8 ≠ F16 &
5754F9 ≠ F10 & F9 ≠ F11 & F9 ≠ F12 & F9 ≠ F13 & F9 ≠ F14 & F9 ≠ F15 & F9 ≠ F16 &
5755F10 ≠ F11 & F10 ≠ F12 & F10 ≠ F13 & F10 ≠ F14 & F10 ≠ F15 & F10 ≠ F16 &
5756F11 ≠ F12 & F11 ≠ F13 & F11 ≠ F14 & F11 ≠ F15 & F11 ≠ F16 &
5757F12 ≠ F13 & F12 ≠ F14 & F12 ≠ F15 & F12 ≠ F16 &
5758F13 ≠ F14 & F13 ≠ F15 & F13 ≠ F16 &
5759F14 ≠ F15 & F14 ≠ F16 &
5760F15 ≠ F16)› 
5761proof -
5762
5763  AOT_have Delta_pos: ‹Δφ → ◇φ› for φ
5764  proof(rule "→I")
5765    AOT_assume ‹Δφ›
5766    AOT_hence ‹□φ ∨ (¬𝒜φ & ◇φ)›
5767      using "≡dfE"[OF necessary_or_contingently_false] by blast
5768    moreover {
5769      AOT_assume ‹□φ›
5770      AOT_hence ‹◇φ›
5771        by (metis "B◇" "T◇" "vdash-properties:10")
5772    }
5773    moreover {
5774      AOT_assume ‹¬𝒜φ & ◇φ›
5775      AOT_hence ‹◇φ›
5776        using "&E" by blast
5777    }
5778    ultimately AOT_show ‹◇φ›
5779      by (metis "∨E"(2) "raa-cor:1") 
5780  qed
5781
5782  AOT_have act_and_not_nec_not_delta: ‹¬Δφ› if ‹𝒜φ› and ‹¬□φ› for φ
5783    using "≡dfE" "&E"(1) "∨E"(2) necessary_or_contingently_false "raa-cor:3" that(1) that(2) by blast
5784  AOT_have act_and_pos_not_not_delta: ‹¬Δφ› if ‹𝒜φ› and ‹◇¬φ› for φ
5785    using "KBasic:11" act_and_not_nec_not_delta "≡E"(2) that(1) that(2) by blast
5786  AOT_have impossible_delta: ‹¬Δφ› if ‹¬◇φ› for φ
5787    using Delta_pos "modus-tollens:1" that by blast
5788  AOT_have not_act_and_pos_delta: ‹Δφ› if ‹¬𝒜φ› and ‹◇φ› for φ
5789    by (meson "≡dfI" "&I" "∨I"(2) necessary_or_contingently_false that(1) that(2))
5790  AOT_have nec_delta: ‹Δφ› if ‹□φ› for φ
5791    using "≡dfI" "∨I"(1) necessary_or_contingently_false that by blast
5792
5793  AOT_obtain a where a_prop: ‹A!a›
5794    using "A-objects"[axiom_inst] "∃E"[rotated] "&E" by blast
5795  AOT_obtain b where b_prop: ‹◇[E!]b & ¬𝒜[E!]b›
5796    using "pos-not-pna:3" using "∃E"[rotated] by blast
5797
5798  AOT_have b_ord: ‹[O!]b›
5799  proof(rule "=dfI"(2)[OF AOT_ordinary])
5800    AOT_show ‹[λx ◇[E!]x]↓› by "cqt:2[lambda]"
5801  next
5802    AOT_show ‹[λx ◇[E!]x]b›
5803    proof (rule "β←C"(1); ("cqt:2[lambda]")?)
5804      AOT_show ‹b↓› by (rule "cqt:2[const_var]"[axiom_inst])
5805      AOT_show ‹◇[E!]b› by (fact b_prop[THEN "&E"(1)])
5806    qed
5807  qed
5808
5809  AOT_have nec_not_L_neg: ‹□¬[L-]x› for x
5810    using "thm-noncont-e-e:2" "contingent-properties:2"[THEN "≡dfE"] "&E"
5811          CBF[THEN "→E"] "∀E" by blast
5812  AOT_have nec_L: ‹□[L]x› for x
5813    using "thm-noncont-e-e:1" "contingent-properties:1"[THEN "≡dfE"]
5814      CBF[THEN "→E"] "∀E" by blast
5815
5816  AOT_have act_ord_b: ‹𝒜[O!]b›
5817    using b_ord "≡E"(1) "oa-facts:7" by blast
5818  AOT_have delta_ord_b: ‹Δ[O!]b›
5819    by (meson "≡dfI" b_ord "∨I"(1) necessary_or_contingently_false "oa-facts:1" "vdash-properties:10")
5820  AOT_have not_act_ord_a: ‹¬𝒜[O!]a›
5821    by (meson a_prop "≡E"(1) "≡E"(3) "oa-contingent:3" "oa-facts:7")
5822  AOT_have not_delta_ord_a: ‹¬Δ[O!]a›
5823    by (metis Delta_pos "≡E"(4) not_act_ord_a "oa-facts:3" "oa-facts:7" "reductio-aa:1" "vdash-properties:10")
5824
5825  AOT_have not_act_abs_b: ‹¬𝒜[A!]b›
5826    by (meson b_ord "≡E"(1) "≡E"(3) "oa-contingent:2" "oa-facts:8")
5827  AOT_have not_delta_abs_b: ‹¬Δ[A!]b›
5828  proof(rule "raa-cor:2")
5829    AOT_assume ‹Δ[A!]b›
5830    AOT_hence ‹◇[A!]b›
5831      by (metis Delta_pos "vdash-properties:10")
5832    AOT_thus ‹[A!]b & ¬[A!]b›
5833      by (metis b_ord "&I" "≡E"(1) "oa-contingent:2" "oa-facts:4" "vdash-properties:10")
5834  qed
5835  AOT_have act_abs_a: ‹𝒜[A!]a›
5836    using a_prop "≡E"(1) "oa-facts:8" by blast
5837  AOT_have delta_abs_a: ‹Δ[A!]a›
5838      by (metis "≡dfI" a_prop "oa-facts:2" "vdash-properties:10" "∨I"(1) necessary_or_contingently_false)
5839
5840  AOT_have not_act_concrete_b: ‹¬𝒜[E!]b›
5841    using b_prop "&E"(2) by blast
5842  AOT_have delta_concrete_b: ‹Δ[E!]b›
5843  proof (rule "≡dfI"[OF necessary_or_contingently_false]; rule "∨I"(2); rule "&I")
5844    AOT_show ‹¬𝒜[E!]b› using b_prop "&E"(2) by blast
5845  next
5846    AOT_show ‹◇[E!]b› using b_prop "&E"(1) by blast
5847  qed
5848  AOT_have not_act_concrete_a: ‹¬𝒜[E!]a›
5849  proof (rule "raa-cor:2")
5850    AOT_assume ‹𝒜[E!]a›
5851    AOT_hence 1: ‹◇[E!]a› by (metis "Act-Sub:3" "vdash-properties:10")
5852    AOT_have ‹[A!]a› by (simp add: a_prop)
5853    AOT_hence ‹[λx ¬◇[E!]x]a›
5854      by (rule "=dfE"(2)[OF AOT_abstract, rotated]) "cqt:2[lambda]"
5855    AOT_hence ‹¬◇[E!]a› using "β→C"(1) by blast
5856    AOT_thus ‹◇[E!]a & ¬◇[E!]a› using 1 "&I" by blast
5857  qed
5858  AOT_have not_delta_concrete_a: ‹¬Δ[E!]a›
5859  proof (rule "raa-cor:2")
5860    AOT_assume ‹Δ[E!]a›
5861    AOT_hence 1: ‹◇[E!]a› by (metis Delta_pos "vdash-properties:10")
5862    AOT_have ‹[A!]a› by (simp add: a_prop)
5863    AOT_hence ‹[λx ¬◇[E!]x]a›
5864      by (rule "=dfE"(2)[OF AOT_abstract, rotated]) "cqt:2[lambda]"
5865    AOT_hence ‹¬◇[E!]a› using "β→C"(1) by blast
5866    AOT_thus ‹◇[E!]a & ¬◇[E!]a› using 1 "&I" by blast
5867  qed
5868
5869  AOT_have not_act_q_zero: ‹¬𝒜q0›
5870    by (meson "log-prop-prop:2" "pos-not-pna:1" q0_def "reductio-aa:1" "rule-id-df:2:a[zero]")
5871  AOT_have delta_q_zero: ‹Δq0›
5872  proof(rule "≡dfI"[OF necessary_or_contingently_false]; rule "∨I"(2); rule "&I")
5873    AOT_show ‹¬𝒜q0› using not_act_q_zero.
5874    AOT_show ‹◇q0› by (meson "&E"(1) q0_prop)
5875  qed
5876  AOT_have act_not_q_zero: ‹𝒜¬q0› using "Act-Basic:1" "∨E"(2) not_act_q_zero by blast
5877  AOT_have not_delta_not_q_zero: ‹¬Δ¬q0›
5878      using "≡dfE" "conventions:5" "Act-Basic:1" act_and_not_nec_not_delta "&E"(1) "∨E"(2) not_act_q_zero q0_prop by blast
5879
5880  AOT_have ‹[L-]↓› by (simp add: "rel-neg-T:3")
5881  moreover AOT_have ‹¬𝒜[L-]b & ¬Δ[L-]b & ¬𝒜[L-]a & ¬Δ[L-]a›
5882  proof (safe intro!: "&I")
5883    AOT_show ‹¬𝒜[L-]b› by (meson "≡E"(1) "logic-actual-nec:1"[axiom_inst] "nec-imp-act" nec_not_L_neg "→E")
5884    AOT_show ‹¬Δ[L-]b› by (meson Delta_pos "KBasic2:1" "≡E"(1) "modus-tollens:1" nec_not_L_neg)
5885    AOT_show ‹¬𝒜[L-]a› by (meson "≡E"(1) "logic-actual-nec:1"[axiom_inst] "nec-imp-act" nec_not_L_neg "→E")
5886    AOT_show ‹¬Δ[L-]a› using Delta_pos "KBasic2:1" "≡E"(1) "modus-tollens:1" nec_not_L_neg by blast
5887  qed
5888  ultimately AOT_obtain F0 where ‹¬𝒜[F0]b & ¬Δ[F0]b & ¬𝒜[F0]a & ¬Δ[F0]a›
5889    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
5890  AOT_hence ‹¬𝒜[F0]b› and ‹¬Δ[F0]b› and ‹¬𝒜[F0]a› and ‹¬Δ[F0]a›
5891    using "&E" by blast+
5892  note props = this
5893
5894  let ?Π = "«[λy [A!]y & q0]»"
5895  AOT_modally_strict {
5896    AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
5897  } note 1 = this
5898  moreover AOT_have‹¬𝒜[«?Π»]b & ¬Δ[«?Π»]b & ¬𝒜[«?Π»]a & Δ[«?Π»]a›
5899  proof(safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
5900    AOT_show ‹¬𝒜([A!]b & q0)›
5901      using "Act-Basic:2" "&E"(1) "≡E"(1) not_act_abs_b "raa-cor:3" by blast
5902  next AOT_show ‹¬Δ([A!]b & q0)›
5903      by (metis Delta_pos "KBasic2:3" "&E"(1) "≡E"(4) not_act_abs_b "oa-facts:4" "oa-facts:8" "raa-cor:3" "vdash-properties:10")
5904  next AOT_show ‹¬𝒜([A!]a & q0)›
5905      using "Act-Basic:2" "&E"(2) "≡E"(1) not_act_q_zero "raa-cor:3" by blast
5906  next AOT_show ‹Δ([A!]a & q0)›
5907    proof (rule not_act_and_pos_delta)
5908      AOT_show ‹¬𝒜([A!]a & q0)›
5909        using "Act-Basic:2" "&E"(2) "≡E"(4) not_act_q_zero "raa-cor:3" by blast
5910    next AOT_show ‹◇([A!]a & q0)›
5911        by (metis "&I" "→E" Delta_pos "KBasic:16" "&E"(1) delta_abs_a "≡E"(1) "oa-facts:6" q0_prop)
5912    qed
5913  qed
5914  ultimately AOT_obtain F1 where ‹¬𝒜[F1]b & ¬Δ[F1]b & ¬𝒜[F1]a & Δ[F1]a›
5915    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
5916  AOT_hence ‹¬𝒜[F1]b› and ‹¬Δ[F1]b› and ‹¬𝒜[F1]a› and ‹Δ[F1]a›
5917    using "&E" by blast+
5918  note props = props this
5919
5920  let ?Π = "«[λy [A!]y & ¬q0]»"
5921  AOT_modally_strict {
5922    AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
5923  } note 1 = this
5924  moreover AOT_have ‹¬𝒜[«?Π»]b & ¬Δ[«?Π»]b & 𝒜[«?Π»]a & ¬Δ[«?Π»]a›
5925  proof(safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
5926    AOT_show ‹¬𝒜([A!]b & ¬q0)›
5927      using "Act-Basic:2" "&E"(1) "≡E"(1) not_act_abs_b "raa-cor:3" by blast
5928  next AOT_show ‹¬Δ([A!]b & ¬q0)›
5929      by (meson "RM◇" Delta_pos "Conjunction Simplification"(1) "≡E"(4) "modus-tollens:1" not_act_abs_b "oa-facts:4" "oa-facts:8")
5930  next AOT_show ‹𝒜([A!]a & ¬q0)›
5931      by (metis "Act-Basic:1" "Act-Basic:2" act_abs_a "&I" "∨E"(2) "≡E"(3) not_act_q_zero "raa-cor:3")
5932  next AOT_show ‹¬Δ([A!]a & ¬q0)›
5933    proof (rule act_and_not_nec_not_delta)
5934      AOT_show ‹𝒜([A!]a & ¬q0)›
5935        by (metis "Act-Basic:1" "Act-Basic:2" act_abs_a "&I" "∨E"(2) "≡E"(3) not_act_q_zero "raa-cor:3")
5936    next
5937      AOT_show ‹¬□([A!]a & ¬q0)›
5938        by (metis "KBasic2:1" "KBasic:3" "&E"(1) "&E"(2) "≡E"(4) q0_prop "raa-cor:3")
5939    qed
5940  qed
5941  ultimately AOT_obtain F2 where ‹¬𝒜[F2]b & ¬Δ[F2]b & 𝒜[F2]a & ¬Δ[F2]a›
5942    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
5943  AOT_hence ‹¬𝒜[F2]b› and ‹¬Δ[F2]b› and ‹𝒜[F2]a› and ‹¬Δ[F2]a›
5944    using "&E" by blast+
5945  note props = props this
5946
5947  AOT_have abstract_prop: ‹¬𝒜[A!]b & ¬Δ[A!]b & 𝒜[A!]a & Δ[A!]a›
5948    using act_abs_a "&I" delta_abs_a not_act_abs_b not_delta_abs_b by presburger
5949  then AOT_obtain F3 where ‹¬𝒜[F3]b & ¬Δ[F3]b & 𝒜[F3]a & Δ[F3]a›
5950    using "∃I"(1)[rotated, THEN "∃E"[rotated]] "oa-exist:2" by fastforce
5951  AOT_hence ‹¬𝒜[F3]b› and ‹¬Δ[F3]b› and ‹𝒜[F3]a› and ‹Δ[F3]a›
5952    using "&E" by blast+
5953  note props = props this
5954
5955  AOT_have ‹¬𝒜[E!]b & Δ[E!]b & ¬𝒜[E!]a & ¬Δ[E!]a›
5956    by (meson "&I" delta_concrete_b not_act_concrete_a not_act_concrete_b not_delta_concrete_a)
5957  then AOT_obtain F4 where ‹¬𝒜[F4]b & Δ[F4]b & ¬𝒜[F4]a & ¬Δ[F4]a›
5958    using "cqt:2[concrete]"[axiom_inst] "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
5959  AOT_hence ‹¬𝒜[F4]b› and ‹Δ[F4]b› and ‹¬𝒜[F4]a› and ‹¬Δ[F4]a›
5960    using "&E" by blast+
5961  note props = props this
5962
5963  AOT_modally_strict {
5964    AOT_have ‹[λy q0]↓› by "cqt:2[lambda]"
5965  } note 1 = this
5966  moreover AOT_have ‹¬𝒜[λy q0]b & Δ[λy q0]b & ¬𝒜[λy q0]a & Δ[λy q0]a›
5967    by (safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
5968       (auto simp: not_act_q_zero delta_q_zero)
5969  ultimately AOT_obtain F5 where ‹¬𝒜[F5]b & Δ[F5]b & ¬𝒜[F5]a & Δ[F5]a›
5970    using "cqt:2[concrete]"[axiom_inst] "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
5971  AOT_hence ‹¬𝒜[F5]b› and ‹Δ[F5]b› and ‹¬𝒜[F5]a› and ‹Δ[F5]a›
5972    using "&E" by blast+
5973  note props = props this
5974
5975  let ?Π = "«[λy [E!]y ∨ ([A!]y & ¬q0)]»"
5976  AOT_modally_strict {
5977    AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
5978  } note 1 = this
5979  moreover AOT_have ‹¬𝒜[«?Π»]b & Δ[«?Π»]b & 𝒜[«?Π»]a & ¬Δ[«?Π»]a›
5980  proof(safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
5981    AOT_have ‹𝒜¬([A!]b & ¬q0)›
5982      by (metis "Act-Basic:1" "Act-Basic:2" abstract_prop "&E"(1) "∨E"(2)
5983                "≡E"(1) "raa-cor:3")
5984    moreover AOT_have ‹¬𝒜[E!]b›
5985      using b_prop "&E"(2) by blast
5986    ultimately AOT_have 2: ‹𝒜(¬[E!]b & ¬([A!]b & ¬q0))›
5987      by (metis "Act-Basic:2" "Act-Sub:1" "&I" "≡E"(3) "raa-cor:1")
5988    AOT_have ‹𝒜¬([E!]b ∨ ([A!]b & ¬q0))›
5989      by (AOT_subst ‹«¬([E!]b ∨ ([A!]b & ¬q0))»› ‹«¬[E!]b & ¬([A!]b & ¬q0)»›)
5990         (auto simp: "oth-class-taut:5:d" 2)
5991    AOT_thus ‹¬𝒜([E!]b ∨ ([A!]b & ¬q0))›
5992      by (metis "¬¬I" "Act-Sub:1" "≡E"(4))
5993  next
5994    AOT_show ‹Δ([E!]b ∨ ([A!]b & ¬q0))›
5995    proof (rule not_act_and_pos_delta)
5996      AOT_show ‹¬𝒜([E!]b ∨ ([A!]b & ¬q0))›
5997        by (metis "Act-Basic:2" "Act-Basic:9" "∨E"(2) "Conjunction Simplification"(1) "≡E"(4) "modus-tollens:1" not_act_abs_b not_act_concrete_b "raa-cor:3")
5998    next
5999      AOT_show ‹◇([E!]b ∨ ([A!]b & ¬q0))›
6000        using "KBasic2:2" b_prop "&E"(1) "∨I"(1) "≡E"(3) "raa-cor:3" by blast
6001    qed
6002  next AOT_show ‹𝒜([E!]a ∨ ([A!]a & ¬q0))›
6003      by (metis "Act-Basic:1" "Act-Basic:2" "Act-Basic:9" act_abs_a "&I" "∨I"(2) "∨E"(2) "≡E"(3) not_act_q_zero "raa-cor:1")
6004  next AOT_show ‹¬Δ([E!]a ∨ ([A!]a & ¬q0))›
6005    proof (rule act_and_not_nec_not_delta)
6006      AOT_show ‹𝒜([E!]a ∨ ([A!]a & ¬q0))›
6007        by (metis "Act-Basic:1" "Act-Basic:2" "Act-Basic:9" act_abs_a "&I" "∨I"(2) "∨E"(2) "≡E"(3) not_act_q_zero "raa-cor:1")
6008    next
6009      AOT_have ‹□¬[E!]a›
6010        by (metis "≡dfI" "conventions:5" "&I" "∨I"(2) necessary_or_contingently_false not_act_concrete_a not_delta_concrete_a "raa-cor:3")
6011      moreover AOT_have ‹◇¬([A!]a & ¬q0)›
6012        by (metis "KBasic2:1" "KBasic:11" "KBasic:3" "&E"(1) "&E"(2) "≡E"(1) q0_prop "raa-cor:3")
6013      ultimately AOT_have ‹◇(¬[E!]a & ¬([A!]a & ¬q0))› by (metis "KBasic:16" "&I" "vdash-properties:10")
6014      AOT_hence ‹◇¬([E!]a ∨ ([A!]a & ¬q0))›
6015        by (metis "RE◇" "≡E"(2) "oth-class-taut:5:d")
6016      AOT_thus ‹¬□([E!]a ∨ ([A!]a & ¬q0))› by (metis "KBasic:12" "≡E"(1) "raa-cor:3")
6017    qed
6018  qed
6019  ultimately AOT_obtain F6 where ‹¬𝒜[F6]b & Δ[F6]b & 𝒜[F6]a & ¬Δ[F6]a›
6020    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6021  AOT_hence ‹¬𝒜[F6]b› and ‹Δ[F6]b› and ‹𝒜[F6]a› and ‹¬Δ[F6]a›
6022    using "&E" by blast+
6023  note props = props this
6024
6025  let ?Π = "«[λy [A!]y ∨ [E!]y]»"
6026  AOT_modally_strict {
6027    AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
6028  } note 1 = this
6029  moreover AOT_have ‹¬𝒜[«?Π»]b & Δ[«?Π»]b & 𝒜[«?Π»]a & Δ[«?Π»]a›
6030  proof(safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6031    AOT_show ‹¬𝒜([A!]b ∨ [E!]b)›
6032      using "Act-Basic:9" "∨E"(2) "≡E"(4) not_act_abs_b not_act_concrete_b "raa-cor:3" by blast
6033  next AOT_show ‹Δ([A!]b ∨ [E!]b)›
6034    proof (rule not_act_and_pos_delta)
6035      AOT_show ‹¬𝒜([A!]b ∨ [E!]b)›
6036        using "Act-Basic:9" "∨E"(2) "≡E"(4) not_act_abs_b not_act_concrete_b "raa-cor:3" by blast
6037    next AOT_show ‹◇([A!]b ∨ [E!]b)›
6038        using "KBasic2:2" b_prop "&E"(1) "∨I"(2) "≡E"(2) by blast
6039    qed
6040  next AOT_show ‹𝒜([A!]a ∨ [E!]a)›
6041      by (meson "Act-Basic:9" act_abs_a "∨I"(1) "≡E"(2))
6042  next AOT_show ‹Δ([A!]a ∨ [E!]a) ›
6043    proof (rule nec_delta)
6044      AOT_show ‹□([A!]a ∨ [E!]a)›
6045        by (metis "KBasic:15" act_abs_a act_and_not_nec_not_delta "Disjunction Addition"(1) delta_abs_a "raa-cor:3" "vdash-properties:10")
6046    qed
6047  qed
6048  ultimately AOT_obtain F7 where ‹¬𝒜[F7]b & Δ[F7]b & 𝒜[F7]a & Δ[F7]a›
6049    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6050  AOT_hence ‹¬𝒜[F7]b› and ‹Δ[F7]b› and ‹𝒜[F7]a› and ‹Δ[F7]a›
6051    using "&E" by blast+
6052  note props = props this
6053
6054  let ?Π = "«[λy [O!]y & ¬[E!]y]»"
6055  AOT_modally_strict {
6056    AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
6057  } note 1 = this
6058  moreover AOT_have ‹𝒜[«?Π»]b & ¬Δ[«?Π»]b & ¬𝒜[«?Π»]a & ¬Δ[«?Π»]a›
6059  proof(safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6060    AOT_show ‹𝒜([O!]b & ¬[E!]b)›
6061      by (metis "Act-Basic:1" "Act-Basic:2" act_ord_b "&I" "∨E"(2) "≡E"(3) not_act_concrete_b "raa-cor:3")
6062  next AOT_show ‹¬Δ([O!]b & ¬[E!]b)›
6063      by (metis (no_types, hide_lams) "conventions:5" "Act-Sub:1" "RM:1" act_and_not_nec_not_delta "act-conj-act:3"
6064                act_ord_b b_prop "&I" "&E"(1) "Conjunction Simplification"(2) "df-rules-formulas[3]"
6065                "≡E"(3) "raa-cor:1" "→E")
6066  next AOT_show ‹¬𝒜([O!]a & ¬[E!]a)›
6067      using "Act-Basic:2" "&E"(1) "≡E"(1) not_act_ord_a "raa-cor:3" by blast
6068  next AOT_have ‹¬◇([O!]a & ¬[E!]a)›
6069      by (metis "KBasic2:3" "&E"(1) "≡E"(4) not_act_ord_a "oa-facts:3" "oa-facts:7" "raa-cor:3" "vdash-properties:10")
6070    AOT_thus ‹¬Δ([O!]a & ¬[E!]a)›
6071      by (rule impossible_delta)
6072  qed      
6073  ultimately AOT_obtain F8 where ‹𝒜[F8]b & ¬Δ[F8]b & ¬𝒜[F8]a & ¬Δ[F8]a›
6074    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6075  AOT_hence ‹𝒜[F8]b› and ‹¬Δ[F8]b› and ‹¬𝒜[F8]a› and ‹¬Δ[F8]a›
6076    using "&E" by blast+
6077  note props = props this
6078
6079  (* TODO_PLM: binary property 9 wrong in PLM *)
6080  let ?Π = "«[λy ¬[E!]y & ([O!]y ∨ q0)]»"
6081  AOT_modally_strict {
6082    AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
6083  } note 1 = this
6084  moreover AOT_have ‹𝒜[«?Π»]b & ¬Δ[«?Π»]b & ¬𝒜[«?Π»]a & Δ[«?Π»]a›
6085  proof(safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6086    AOT_show ‹𝒜(¬[E!]b & ([O!]b ∨ q0))›
6087      by (metis "Act-Basic:1" "Act-Basic:2" "Act-Basic:9" act_ord_b "&I" "∨I"(1)
6088                "∨E"(2) "≡E"(3) not_act_concrete_b "raa-cor:1")
6089  next AOT_show ‹¬Δ(¬[E!]b & ([O!]b ∨ q0))›
6090    proof (rule act_and_pos_not_not_delta)
6091      AOT_show ‹𝒜(¬[E!]b & ([O!]b ∨ q0))›
6092        by (metis "Act-Basic:1" "Act-Basic:2" "Act-Basic:9" act_ord_b "&I" "∨I"(1)
6093                  "∨E"(2) "≡E"(3) not_act_concrete_b "raa-cor:1")
6094    next
6095      AOT_show ‹◇¬(¬[E!]b & ([O!]b ∨ q0))›
6096      proof (AOT_subst ‹«¬(¬[E!]b & ([O!]b ∨ q0))»› ‹«[E!]b ∨ ¬([O!]b ∨ q0)»›)
6097        AOT_modally_strict {
6098          AOT_show ‹¬(¬[E!]b & ([O!]b ∨ q0)) ≡ [E!]b ∨ ¬([O!]b ∨ q0)›
6099            by (metis "&I" "&E"(1) "&E"(2) "∨I"(1) "∨I"(2) "∨E"(2) "deduction-theorem" "≡I" "reductio-aa:1")
6100        }
6101      next
6102        AOT_show ‹◇([E!]b ∨ ¬([O!]b ∨ q0))›
6103          using "KBasic2:2" b_prop "&E"(1) "∨I"(1) "≡E"(3) "raa-cor:3" by blast
6104       qed
6105     qed
6106   next
6107     AOT_show ‹¬𝒜(¬[E!]a & ([O!]a ∨ q0))›
6108       using "Act-Basic:2" "Act-Basic:9" "&E"(2) "∨E"(3) "≡E"(1) not_act_ord_a not_act_q_zero "reductio-aa:2" by blast
6109   next
6110     AOT_show ‹Δ(¬[E!]a & ([O!]a ∨ q0))›
6111     proof (rule not_act_and_pos_delta)
6112       AOT_show ‹¬𝒜(¬[E!]a & ([O!]a ∨ q0))›
6113         by (metis "Act-Basic:2" "Act-Basic:9" "&E"(2) "∨E"(3) "≡E"(1) not_act_ord_a not_act_q_zero "reductio-aa:2")
6114     next
6115       AOT_have ‹□¬[E!]a›
6116         using "KBasic2:1" "≡E"(2) not_act_and_pos_delta not_act_concrete_a not_delta_concrete_a "raa-cor:5" by blast
6117       moreover AOT_have ‹◇([O!]a ∨ q0)›
6118         by (metis "KBasic2:2" "&E"(1) "∨I"(2) "≡E"(3) q0_prop "raa-cor:3")
6119       ultimately AOT_show ‹◇(¬[E!]a & ([O!]a ∨ q0))›
6120         by (metis "KBasic:16" "&I" "vdash-properties:10")
6121     qed
6122   qed
6123  ultimately AOT_obtain F9 where ‹𝒜[F9]b & ¬Δ[F9]b & ¬𝒜[F9]a & Δ[F9]a›
6124    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6125  AOT_hence ‹𝒜[F9]b› and ‹¬Δ[F9]b› and ‹¬𝒜[F9]a› and ‹Δ[F9]a›
6126    using "&E" by blast+
6127  note props = props this
6128
6129  AOT_modally_strict {
6130    AOT_have ‹[λy ¬q0]↓› by "cqt:2[lambda]"
6131  } note 1 = this
6132  moreover AOT_have ‹𝒜[λy ¬q0]b & ¬Δ[λy ¬q0]b & 𝒜[λy ¬q0]a & ¬Δ[λy ¬q0]a›
6133    by (safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1]; auto simp: act_not_q_zero not_delta_not_q_zero)
6134  ultimately AOT_obtain F10 where ‹𝒜[F10]b & ¬Δ[F10]b & 𝒜[F10]a & ¬Δ[F10]a›
6135    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6136  AOT_hence ‹𝒜[F10]b› and ‹¬Δ[F10]b› and ‹𝒜[F10]a› and ‹¬Δ[F10]a›
6137    using "&E" by blast+
6138  note props = props this
6139
6140  AOT_modally_strict {
6141    AOT_have ‹[λy ¬[E!]y]↓› by "cqt:2[lambda]"
6142  } note 1 = this
6143  moreover AOT_have ‹𝒜[λy ¬[E!]y]b & ¬Δ[λy ¬[E!]y]b & 𝒜[λy ¬[E!]y]a & Δ[λy ¬[E!]y]a›
6144  proof (safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6145    AOT_show ‹𝒜¬[E!]b›
6146      using "Act-Basic:1" "∨E"(2) not_act_concrete_b by blast
6147  next AOT_show ‹¬Δ¬[E!]b›
6148      using "≡dfE" "conventions:5" "Act-Basic:1" act_and_not_nec_not_delta b_prop "&E"(1) "∨E"(2) not_act_concrete_b by blast
6149  next AOT_show ‹𝒜¬[E!]a›
6150      using "Act-Basic:1" "∨E"(2) not_act_concrete_a by blast
6151  next AOT_show ‹Δ¬[E!]a›
6152      using "KBasic2:1" "≡E"(2) nec_delta not_act_and_pos_delta not_act_concrete_a not_delta_concrete_a "reductio-aa:1" by blast
6153  qed
6154  ultimately AOT_obtain F11 where ‹𝒜[F11]b & ¬Δ[F11]b & 𝒜[F11]a & Δ[F11]a›
6155    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6156  AOT_hence ‹𝒜[F11]b› and ‹¬Δ[F11]b› and ‹𝒜[F11]a› and ‹Δ[F11]a›
6157    using "&E" by blast+
6158  note props = props this
6159
6160  AOT_have ‹𝒜[O!]b & Δ[O!]b & ¬𝒜[O!]a & ¬Δ[O!]a›
6161    by (simp add: act_ord_b "&I" delta_ord_b not_act_ord_a not_delta_ord_a)
6162  then AOT_obtain F12 where ‹𝒜[F12]b & Δ[F12]b & ¬𝒜[F12]a & ¬Δ[F12]a›
6163    using "oa-exist:1" "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6164  AOT_hence ‹𝒜[F12]b› and ‹Δ[F12]b› and ‹¬𝒜[F12]a› and ‹¬Δ[F12]a›
6165    using "&E" by blast+
6166  note props = props this
6167
6168  let ?Π = "«[λy [O!]y ∨ q0]»"
6169  AOT_modally_strict {
6170    AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
6171  } note 1 = this
6172  moreover AOT_have ‹𝒜[«?Π»]b & Δ[«?Π»]b & ¬𝒜[«?Π»]a & Δ[«?Π»]a›
6173  proof (safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6174    AOT_show ‹𝒜([O!]b ∨ q0)›
6175      by (meson "Act-Basic:9" act_ord_b "∨I"(1) "≡E"(2))
6176  next AOT_show ‹Δ([O!]b ∨ q0)›
6177      by (meson "KBasic:15" b_ord "∨I"(1) nec_delta "oa-facts:1" "vdash-properties:10")
6178  next AOT_show ‹¬𝒜([O!]a ∨ q0)›
6179      using "Act-Basic:9" "∨E"(2) "≡E"(4) not_act_ord_a not_act_q_zero "raa-cor:3" by blast
6180  next AOT_show ‹Δ([O!]a ∨ q0)›
6181    proof (rule not_act_and_pos_delta)
6182      AOT_show ‹¬𝒜([O!]a ∨ q0)›
6183        using "Act-Basic:9" "∨E"(2) "≡E"(4) not_act_ord_a not_act_q_zero "raa-cor:3" by blast
6184    next AOT_show ‹◇([O!]a ∨ q0)›
6185        using "KBasic2:2" "&E"(1) "∨I"(2) "≡E"(2) q0_prop by blast
6186    qed
6187  qed
6188  ultimately AOT_obtain F13 where ‹𝒜[F13]b & Δ[F13]b & ¬𝒜[F13]a & Δ[F13]a›
6189    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6190  AOT_hence ‹𝒜[F13]b› and ‹Δ[F13]b› and ‹¬𝒜[F13]a› and ‹Δ[F13]a›
6191    using "&E" by blast+
6192  note props = props this
6193
6194  let ?Π = "«[λy [O!]y ∨ ¬q0]»"
6195  AOT_modally_strict {
6196     AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
6197  } note 1 = this
6198  moreover AOT_have ‹𝒜[«?Π»]b & Δ[«?Π»]b & 𝒜[«?Π»]a & ¬Δ[«?Π»]a›
6199  proof (safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6200    AOT_show ‹𝒜([O!]b ∨ ¬q0)›
6201      by (meson "Act-Basic:9" act_not_q_zero "∨I"(2) "≡E"(2))
6202  next AOT_show ‹Δ([O!]b ∨ ¬q0)›
6203      by (meson "KBasic:15" b_ord "∨I"(1) nec_delta "oa-facts:1" "vdash-properties:10")
6204  next AOT_show ‹𝒜([O!]a ∨ ¬q0)›
6205      by (meson "Act-Basic:9" act_not_q_zero "∨I"(2) "≡E"(2))
6206  next AOT_show ‹¬Δ([O!]a ∨ ¬q0)›
6207    proof(rule act_and_pos_not_not_delta)
6208      AOT_show ‹𝒜([O!]a ∨ ¬q0)›
6209        by (meson "Act-Basic:9" act_not_q_zero "∨I"(2) "≡E"(2))
6210    next
6211      AOT_have ‹□¬[O!]a›
6212        using "KBasic2:1" "≡E"(2) not_act_and_pos_delta not_act_ord_a not_delta_ord_a "raa-cor:6" by blast
6213      moreover AOT_have ‹◇q0›
6214        by (meson "&E"(1) q0_prop)
6215      ultimately AOT_have 2: ‹◇(¬[O!]a & q0)›
6216         by (metis "KBasic:16" "&I" "vdash-properties:10")
6217      AOT_show ‹◇¬([O!]a ∨ ¬q0)›
6218      proof (AOT_subst_rev ‹«¬[O!]a & q0»› ‹«¬([O!]a ∨ ¬q0)»›)
6219        AOT_modally_strict {
6220          AOT_show ‹¬[O!]a & q0 ≡ ¬([O!]a ∨ ¬q0)›
6221            by (metis "&I" "&E"(1) "&E"(2) "∨I"(1) "∨I"(2)
6222                      "∨E"(3) "deduction-theorem" "≡I" "raa-cor:3")
6223        }
6224      next
6225        AOT_show ‹◇(¬[O!]a & q0)›
6226          using "2" by blast
6227      qed
6228    qed
6229  qed
6230  ultimately AOT_obtain F14 where ‹𝒜[F14]b & Δ[F14]b & 𝒜[F14]a & ¬Δ[F14]a›
6231    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6232  AOT_hence ‹𝒜[F14]b› and ‹Δ[F14]b› and ‹𝒜[F14]a› and ‹¬Δ[F14]a›
6233    using "&E" by blast+
6234  note props = props this
6235
6236  AOT_have ‹[L]↓›
6237    by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
6238  moreover AOT_have ‹𝒜[L]b & Δ[L]b & 𝒜[L]a & Δ[L]a›
6239  proof (safe intro!: "&I")
6240    AOT_show ‹𝒜[L]b›
6241      by (meson nec_L "nec-imp-act" "vdash-properties:10")
6242    next AOT_show ‹Δ[L]b› using nec_L nec_delta by blast
6243    next AOT_show ‹𝒜[L]a› by (meson nec_L "nec-imp-act" "vdash-properties:10")
6244    next AOT_show ‹Δ[L]a› using nec_L nec_delta by blast
6245  qed
6246  ultimately AOT_obtain F15 where ‹𝒜[F15]b & Δ[F15]b & 𝒜[F15]a & Δ[F15]a›
6247    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6248  AOT_hence ‹𝒜[F15]b› and ‹Δ[F15]b› and ‹𝒜[F15]a› and ‹Δ[F15]a›
6249    using "&E" by blast+
6250  note props = props this
6251
6252  show ?thesis
6253    by (rule "∃I"(2)[where β=F0]; rule "∃I"(2)[where β=F1]; rule "∃I"(2)[where β=F2];
6254           rule "∃I"(2)[where β=F3]; rule "∃I"(2)[where β=F4]; rule "∃I"(2)[where β=F5];
6255           rule "∃I"(2)[where β=F6]; rule "∃I"(2)[where β=F7]; rule "∃I"(2)[where β=F8];
6256           rule "∃I"(2)[where β=F9]; rule "∃I"(2)[where β=F10]; rule "∃I"(2)[where β=F11];
6257           rule "∃I"(2)[where β=F12]; rule "∃I"(2)[where β=F13]; rule "∃I"(2)[where β=F14];
6258           rule "∃I"(2)[where β=F15]; safe intro!: "&I")
6259       (match conclusion in "[?v ⊨ [F] ≠ [G]]" for F G ⇒ ‹
6260        match props in A: "[?v ⊨ ¬φ{F}]" for φ ⇒ ‹
6261        match (φ) in "λa . ?p" ⇒ ‹fail› ¦ "λa . a" ⇒ ‹fail› ¦ _ ⇒ ‹
6262        match props in B: "[?v ⊨ φ{G}]" ⇒ ‹
6263        fact "pos-not-equiv-ne:4"[where F=F and G=G and φ=φ, THEN "→E",
6264                                OF "oth-class-taut:4:h"[THEN "≡E"(2)],
6265                                OF "Disjunction Addition"(2)[THEN "→E"],
6266                                OF "&I", OF A, OF B]››››)+
6267qed
6268
6269AOT_theorem "o-objects-exist:1": ‹□∃x O!x›
6270proof(rule RN)
6271  AOT_modally_strict {
6272    AOT_obtain a where ‹◇(E!a & ¬𝒜[E!]a)›
6273      using "∃E"[rotated, OF "qml:4"[axiom_inst, THEN "BF◇"[THEN "→E"]]] by blast
6274    AOT_hence 1: ‹◇E!a› by (metis "KBasic2:3" "&E"(1) "→E")
6275    AOT_have ‹[λx ◇[E!]x]a›
6276    proof (rule "β←C"(1); "cqt:2[lambda]"?)
6277      AOT_show ‹a↓› using "cqt:2[const_var]"[axiom_inst] by blast
6278    next
6279      AOT_show ‹◇E!a› by (fact 1)
6280    qed
6281    AOT_hence ‹O!a› by (rule "=dfI"(2)[OF AOT_ordinary, rotated]) "cqt:2[lambda]"
6282    AOT_thus ‹∃x [O!]x› by (rule "∃I")
6283  }
6284qed
6285
6286AOT_theorem "o-objects-exist:2": ‹□∃x A!x›
6287proof (rule RN)
6288  AOT_modally_strict {
6289    AOT_obtain a where ‹[A!]a›
6290      using "A-objects"[axiom_inst] "∃E"[rotated] "&E" by blast
6291    AOT_thus ‹∃x A!x› using "∃I" by blast
6292  }
6293qed
6294
6295AOT_theorem "o-objects-exist:3": ‹□¬∀x O!x›
6296  by (rule RN) (metis (no_types, hide_lams) "∃E" "cqt-orig:1[const_var]" "≡E"(4) "modus-tollens:1" "o-objects-exist:2" "oa-contingent:2" "qml:2"[axiom_inst] "reductio-aa:2")
6297
6298AOT_theorem "o-objects-exist:4": ‹□¬∀x A!x›
6299  by (rule RN) (metis (mono_tags, hide_lams) "∃E" "cqt-orig:1[const_var]" "≡E"(1) "modus-tollens:1" "o-objects-exist:1" "oa-contingent:2" "qml:2"[axiom_inst] "→E")
6300
6301AOT_theorem "o-objects-exist:5": ‹□¬∀x E!x›
6302proof (rule RN; rule "raa-cor:2")
6303  AOT_modally_strict {
6304    AOT_assume ‹∀x E!x›
6305    moreover AOT_obtain a where abs: ‹A!a›
6306      using "o-objects-exist:2"[THEN "qml:2"[axiom_inst, THEN "→E"]] "∃E"[rotated] by blast
6307    ultimately AOT_have ‹E!a› using "∀E" by blast
6308    AOT_hence 1: ‹◇E!a› by (metis "T◇" "→E")
6309    AOT_have ‹[λy ◇E!y]a›
6310    proof (rule "β←C"(1); "cqt:2[lambda]"?)
6311      AOT_show ‹a↓› using "cqt:2[const_var]"[axiom_inst].
6312    next
6313      AOT_show ‹◇E!a› by (fact 1)
6314    qed
6315    AOT_hence ‹O!a›
6316      by (rule "=dfI"(2)[OF AOT_ordinary, rotated]) "cqt:2[lambda]"
6317    AOT_hence ‹¬A!a› by (metis "≡E"(1) "oa-contingent:2") 
6318    AOT_thus ‹p & ¬p› for p using abs by (metis "raa-cor:3")
6319  }
6320qed
6321
6322AOT_theorem partition: ‹¬∃x (O!x & A!x)›
6323proof(rule "raa-cor:2")
6324  AOT_assume ‹∃x (O!x & A!x)›
6325  then AOT_obtain a where ‹O!a & A!a› using "∃E"[rotated] by blast
6326  AOT_thus ‹p & ¬p› for p by (metis "&E"(1) "Conjunction Simplification"(2) "≡E"(1) "modus-tollens:1" "oa-contingent:2" "raa-cor:3")
6327qed
6328
6329AOT_define eq_E :: ‹Π› ("'(=E')") "=E": ‹(=E) =df [λxy O!x & O!y & □∀F ([F]x ≡ [F]y)]›
6330
6331syntax "_AOT_eq_E_infix" :: ‹τ ⇒ τ ⇒ φ› (infixl "=E" 50)
6332translations
6333  "_AOT_eq_E_infix κ κ'" == "CONST AOT_exe (CONST eq_E) (CONST Pair κ κ')"
6334(* TODO: try to replace by a simple translations pattern *)
6335print_translation‹
6336AOT_syntax_print_translations
6337[(const_syntax‹AOT_exe›, fn ctxt => fn [
6338  Const ("constAOT_PLM.eq_E", _),
6339  Const (const_syntax‹Pair›, _) $ lhs $ rhs
6340] => Const (syntax_const‹_AOT_eq_E_infix›, dummyT) $ lhs $ rhs)]›
6341
6342text‹Note: Not explicitly mentioned as theorem in PLM.›
6343AOT_theorem "=E[denotes]": ‹[(=E)]↓›
6344  by (rule "=dfI"(2)[OF "=E"]) "cqt:2[lambda]"+
6345
6346AOT_theorem "=E-simple:1": ‹x =E y ≡ (O!x & O!y & □∀F ([F]x ≡ [F]y))›
6347proof -
6348  (* TODO: rethink the product hacks *)
6349  AOT_have 0: ‹«(AOT_term_of_var x,AOT_term_of_var y)»↓›
6350    by (simp add: "&I" "cqt:2[const_var]" prod_denotesI "vdash-properties:1[2]")
6351  AOT_have 1: ‹[λxy [O!]x & [O!]y & □∀F ([F]x ≡ [F]y)]↓› by "cqt:2[lambda]"
6352  show ?thesis apply (rule "=dfI"(2)[OF "=E"]; "cqt:2[lambda]"?)
6353    using "beta-C-meta"[THEN "→E", OF 1, unvarify ν1νn, of "(AOT_term_of_var x,AOT_term_of_var y)", OF 0]
6354    by fast
6355qed
6356
6357AOT_theorem "=E-simple:2": ‹x =E y → x = y›
6358proof (rule "→I")
6359  AOT_assume ‹x =E y›
6360  AOT_hence ‹O!x & O!y & □∀F ([F]x ≡ [F]y)› using "=E-simple:1"[THEN "≡E"(1)] by blast
6361  AOT_thus ‹x = y›
6362    using "≡dfI"[OF "identity:1"] "∨I" by blast
6363qed
6364
6365AOT_theorem "id-nec3:1": ‹x =E y ≡ □(x =E y)›
6366proof (rule "≡I"; rule "→I")
6367  AOT_assume ‹x =E y›
6368  AOT_hence ‹O!x & O!y & □∀F ([F]x ≡ [F]y)›
6369    using "=E-simple:1" "≡E" by blast
6370  AOT_hence ‹□O!x & □O!y & □□∀F ([F]x ≡ [F]y)›
6371    by (metis "S5Basic:6" "&I" "&E"(1) "&E"(2) "≡E"(4) "oa-facts:1" "raa-cor:3" "vdash-properties:10")
6372  AOT_hence 1: ‹□(O!x & O!y & □∀F ([F]x ≡ [F]y))›
6373    by (metis "&E"(1) "&E"(2) "≡E"(2) "KBasic:3" "&I")
6374  AOT_show ‹□(x =E y)›
6375    apply (AOT_subst ‹«x =E y»› ‹«O!x & O!y & □∀F ([F]x ≡ [F]y)»›)
6376     using "=E-simple:1" apply presburger
6377    by (simp add: "1")
6378next
6379  AOT_assume ‹□(x =E y)›
6380  AOT_thus ‹x =E y› using "qml:2"[axiom_inst, THEN "→E"] by blast
6381qed
6382
6383AOT_theorem "id-nec3:2": ‹◇(x =E y) ≡ x =E y›
6384  by (meson "RE◇" "S5Basic:2" "id-nec3:1" "≡E"(1) "≡E"(5) "Commutativity of ≡")
6385
6386AOT_theorem "id-nec3:3": ‹◇(x =E y) ≡ □(x =E y)›
6387  by (meson "id-nec3:1" "id-nec3:2" "≡E"(5))
6388
6389syntax "_AOT_non_eq_E" :: ‹Π› ("'(≠E')")
6390translations
6391  (Π) "(≠E)" == (Π) "(=E)-"
6392syntax "_AOT_non_eq_E_infix" :: ‹τ ⇒ τ ⇒ φ› (infixl "≠E" 50)
6393translations
6394 "_AOT_non_eq_E_infix κ κ'" == "CONST AOT_exe (CONST relation_negation (CONST eq_E)) (CONST Pair κ κ')"
6395(* TODO: try replacing be a simple translations pattern *)
6396print_translation‹
6397AOT_syntax_print_translations
6398[(const_syntax‹AOT_exe›, fn ctxt => fn [
6399  Const (const_syntax‹relation_negation›, _) $ Const ("constAOT_PLM.eq_E", _),
6400  Const (const_syntax‹Pair›, _) $ lhs $ rhs
6401] => Const (syntax_const‹_AOT_non_eq_E_infix›, dummyT) $ lhs $ rhs)]›
6402AOT_theorem "thm-neg=E": ‹x ≠E y ≡ ¬(x =E y)›
6403proof -
6404  (* TODO: rethink the product hacks *)
6405  AOT_have 0: ‹«(AOT_term_of_var x,AOT_term_of_var y)»↓›
6406    by (simp add: "&I" "cqt:2[const_var]" prod_denotesI "vdash-properties:1[2]")
6407  AOT_have θ: ‹[λx1...x2 ¬(=E)x1...x2]↓› by "cqt:2[lambda]" (* TODO_PLM: convoluted proof in PLM; TODO: product hack *)
6408  AOT_have ‹x ≠E y ≡ [λx1...x2 ¬(=E)x1...x2]xy›
6409    by (rule "=dfI"(1)[OF "df-relation-negation", OF θ])
6410       (meson "oth-class-taut:3:a")
6411  also AOT_have ‹… ≡ ¬(=E)xy›
6412    apply (rule "beta-C-meta"[THEN "→E", unvarify ν1νn])
6413     apply "cqt:2[lambda]"
6414    by (fact 0)
6415  finally show ?thesis.
6416qed
6417
6418AOT_theorem "id-nec4:1": ‹x ≠E y ≡ □(x ≠E y)›
6419proof -
6420  AOT_have ‹x ≠E y ≡ ¬(x =E y)› using "thm-neg=E".
6421  also AOT_have ‹… ≡ ¬◇(x =E y)›
6422    by (meson "id-nec3:2" "≡E"(1) "Commutativity of ≡" "oth-class-taut:4:b")
6423  also AOT_have ‹… ≡ □¬(x =E y)›
6424    by (meson "KBasic2:1" "≡E"(2) "Commutativity of ≡")
6425  also AOT_have ‹… ≡ □(x ≠E y)›
6426    by (AOT_subst_rev "«x ≠E y»" "«¬(x =E y)»")
6427       (auto simp: "thm-neg=E" "oth-class-taut:3:a")
6428  finally show ?thesis.
6429qed
6430
6431AOT_theorem "id-nec4:2": ‹◇(x ≠E y) ≡ (x ≠E y)›
6432  by (meson "RE◇" "S5Basic:2" "id-nec4:1" "≡E"(2) "≡E"(5) "Commutativity of ≡")
6433
6434AOT_theorem "id-nec4:3": ‹◇(x ≠E y) ≡ □(x ≠E y)›
6435  by (meson "id-nec4:1" "id-nec4:2" "≡E"(5))
6436
6437AOT_theorem "id-act2:1": ‹x =E y ≡ 𝒜x =E y›
6438  by (meson "Act-Basic:5" "Act-Sub:2" "RA[2]" "id-nec3:2" "≡E"(1) "≡E"(6))
6439AOT_theorem "id-act2:2": ‹x ≠E y ≡ 𝒜x ≠E y›
6440  by (meson "Act-Basic:5" "Act-Sub:2" "RA[2]" "id-nec4:2" "≡E"(1) "≡E"(6))
6441
6442AOT_theorem "ord=Eequiv:1": ‹O!x → x =E x›
6443proof (rule "→I")
6444  AOT_assume 1: ‹O!x›
6445  AOT_show ‹x =E x›
6446    apply (rule "=dfI"(2)[OF "=E"]) apply "cqt:2[lambda]"
6447    apply (rule "β←C"(1))
6448      apply "cqt:2[lambda]"
6449     apply (simp add: "&I" "cqt:2[const_var]" prod_denotesI "vdash-properties:1[2]")
6450    by (simp add: "1" RN "&I" "oth-class-taut:3:a" "universal-cor")
6451qed
6452
6453AOT_theorem "ord=Eequiv:2": ‹x =E y → y =E x›
6454proof(rule CP)
6455  AOT_assume 1: ‹x =E y›
6456  AOT_hence 2: ‹x = y› by (metis "=E-simple:2" "vdash-properties:10") 
6457  AOT_have ‹O!x› using 1 by (meson "&E"(1) "=E-simple:1" "≡E"(1))
6458  AOT_hence ‹x =E x› using "ord=Eequiv:1" "→E" by blast
6459  AOT_thus ‹y =E x› using "rule=E"[rotated, OF 2] by fast
6460qed
6461
6462AOT_theorem "ord=Eequiv:3": ‹(x =E y & y =E z) → x =E z›
6463proof (rule CP)
6464  AOT_assume 1: ‹x =E y & y =E z›
6465  AOT_hence ‹x = y & y = z›
6466    by (metis "&I" "&E"(1) "&E"(2) "=E-simple:2" "vdash-properties:6")
6467  AOT_hence ‹x = z› by (metis "id-eq:3" "vdash-properties:6")
6468  moreover AOT_have ‹x =E x›
6469    using 1[THEN "&E"(1)] "&E"(1) "=E-simple:1" "≡E"(1) "ord=Eequiv:1" "→E" by blast
6470  ultimately AOT_show ‹x =E z›
6471    using "rule=E" by fast
6472qed
6473
6474AOT_theorem "ord-=E=:1": ‹(O!x ∨ O!y) → □(x = y ≡ x =E y)›
6475proof(rule CP)
6476  AOT_assume ‹O!x ∨ O!y›
6477  moreover {
6478    AOT_assume ‹O!x›
6479    AOT_hence ‹□O!x› by (metis "oa-facts:1" "vdash-properties:10")
6480    moreover {
6481      AOT_modally_strict {
6482        AOT_have ‹O!x → (x = y ≡ x =E y)›
6483        proof (rule "→I"; rule "≡I"; rule "→I")
6484          AOT_assume ‹O!x›
6485          AOT_hence ‹x =E x› by (metis "ord=Eequiv:1" "→E")
6486          moreover AOT_assume ‹x = y›
6487          ultimately AOT_show ‹x =E y› using "rule=E" by fast
6488        next
6489          AOT_assume ‹x =E y›
6490          AOT_thus ‹x = y› by (metis "=E-simple:2" "→E")
6491        qed
6492      }
6493      AOT_hence ‹□O!x → □(x = y ≡ x =E y)› by (metis "RM:1")
6494    }
6495    ultimately AOT_have ‹□(x = y ≡ x =E y)› using "→E" by blast
6496  }
6497  moreover {
6498    AOT_assume ‹O!y›
6499    AOT_hence ‹□O!y› by (metis "oa-facts:1" "vdash-properties:10")
6500    moreover {
6501      AOT_modally_strict {
6502        AOT_have ‹O!y → (x = y ≡ x =E y)›
6503        proof (rule "→I"; rule "≡I"; rule "→I")
6504          AOT_assume ‹O!y›
6505          AOT_hence ‹y =E y› by (metis "ord=Eequiv:1" "→E")
6506          moreover AOT_assume ‹x = y›
6507          ultimately AOT_show ‹x =E y› using "rule=E" id_sym by fast
6508        next
6509          AOT_assume ‹x =E y›
6510          AOT_thus ‹x = y› by (metis "=E-simple:2" "→E")
6511        qed
6512      }
6513      AOT_hence ‹□O!y → □(x = y ≡ x =E y)› by (metis "RM:1")
6514    }
6515    ultimately AOT_have ‹□(x = y ≡ x =E y)› using "→E" by blast
6516  }
6517  ultimately AOT_show ‹□(x = y ≡ x =E y)› by (metis "∨E"(3) "raa-cor:1")
6518qed
6519
6520AOT_theorem "ord-=E=:2": ‹O!y → [λx x = y]↓›
6521proof (rule "→I"; rule "safe-ext"[axiom_inst, THEN "→E"]; rule "&I")
6522  AOT_show ‹[λx x =E y]↓› by "cqt:2[lambda]"
6523next
6524  AOT_assume ‹O!y›
6525  AOT_hence 1: ‹□(x = y ≡ x =E y)› for x using "ord-=E=:1" "→E" "∨I" by blast
6526  AOT_have ‹□(x =E y ≡ x = y)› for x
6527    by (AOT_subst ‹«x =E y ≡ x = y»› ‹«x = y ≡ x =E y»›)
6528       (auto simp add: "Commutativity of ≡" 1)
6529  AOT_hence ‹∀x □(x =E y ≡ x = y)› by (rule GEN)
6530  AOT_thus ‹□∀x (x =E y ≡ x = y)› by (rule BF[THEN "→E"])
6531qed
6532
6533
6534AOT_theorem "ord-=E=:3": ‹[λxy O!x & O!y & x = y]↓›
6535proof (rule "safe-ext[2]"[axiom_inst, THEN "→E"]; rule "&I")
6536  AOT_show ‹[λxy O!x & O!y & x =E y]↓› by "cqt:2[lambda]"
6537next
6538  AOT_show ‹□∀x∀y ([O!]x & [O!]y & x =E y ≡ [O!]x & [O!]y & x = y)›
6539  proof (rule RN; rule GEN; rule GEN; rule "≡I"; rule "→I")
6540    AOT_modally_strict {
6541      AOT_show ‹[O!]x & [O!]y & x = y› if ‹[O!]x & [O!]y & x =E y› for x y
6542        by (metis "&I" "&E"(1) "Conjunction Simplification"(2) "=E-simple:2"
6543                  "modus-tollens:1" "raa-cor:1" that)
6544    }
6545  next
6546    AOT_modally_strict {
6547      AOT_show ‹[O!]x & [O!]y & x =E y› if ‹[O!]x & [O!]y & x = y› for x y
6548        apply(safe intro!: "&I")
6549          apply (metis that[THEN "&E"(1), THEN "&E"(1)])
6550         apply (metis that[THEN "&E"(1), THEN "&E"(2)])
6551        using "rule=E"[rotated, OF that[THEN "&E"(2)]]
6552              "ord=Eequiv:1"[THEN "→E", OF that[THEN "&E"(1), THEN "&E"(1)]] by fast
6553    }
6554  qed
6555qed
6556
6557AOT_theorem "ind-nec": ‹∀F ([F]x ≡ [F]y) → □∀F ([F]x ≡ [F]y)›
6558proof(rule "→I")
6559  AOT_assume ‹∀F ([F]x ≡ [F]y)›
6560  moreover AOT_have ‹[λx □∀F ([F]x ≡ [F]y)]↓› by "cqt:2[lambda]"
6561  ultimately AOT_have ‹[λx □∀F ([F]x ≡ [F]y)]x ≡ [λx □∀F ([F]x ≡ [F]y)]y›
6562    using "∀E" by blast
6563  moreover AOT_have ‹[λx □∀F ([F]x ≡ [F]y)]y›
6564    apply (rule "β←C"(1))
6565      apply "cqt:2[lambda]"
6566     apply (fact "cqt:2[const_var]"[axiom_inst])
6567    by (simp add: RN GEN "oth-class-taut:3:a")
6568  ultimately AOT_have ‹[λx □∀F ([F]x ≡ [F]y)]x› using "≡E" by blast
6569  AOT_thus ‹□∀F ([F]x ≡ [F]y)›
6570    using "β→C"(1) by blast
6571qed
6572
6573AOT_theorem "ord=E:1": ‹(O!x & O!y) → (∀F ([F]x ≡ [F]y) → x =E y)›
6574proof (rule "→I"; rule "→I")
6575  AOT_assume ‹∀F ([F]x ≡ [F]y)›
6576  AOT_hence ‹□∀F ([F]x ≡ [F]y)›
6577    using "ind-nec"[THEN "→E"] by blast
6578  moreover AOT_assume ‹O!x & O!y›
6579  ultimately AOT_have ‹O!x & O!y & □∀F ([F]x ≡ [F]y)›
6580    using "&I" by blast
6581  AOT_thus ‹x =E y› using "=E-simple:1"[THEN "≡E"(2)] by blast
6582qed
6583
6584AOT_theorem "ord=E:2": ‹(O!x & O!y) → (∀F ([F]x ≡ [F]y) → x = y)›
6585proof (rule "→I"; rule "→I")
6586  AOT_assume ‹O!x & O!y›
6587  moreover AOT_assume ‹∀F ([F]x ≡ [F]y)›
6588  ultimately AOT_have ‹x =E y›
6589    using "ord=E:1" "→E" by blast
6590  AOT_thus ‹x = y› using "=E-simple:2"[THEN "→E"] by blast
6591qed
6592
6593AOT_theorem "ord=E2:1": ‹(O!x & O!y) → (x ≠ y ≡ [λz z =E x] ≠ [λz z =E y])›
6594proof (rule "→I"; rule "≡I"; rule "→I"; rule "≡dfI"[OF "=-infix"]; rule "raa-cor:2")
6595  AOT_assume 0: ‹O!x & O!y›
6596  AOT_assume ‹x ≠ y›
6597  AOT_hence 1: ‹¬(x = y)› using "≡dfE"[OF "=-infix"] by blast
6598  AOT_assume ‹[λz z =E x] = [λz z =E y]›
6599  moreover AOT_have ‹[λz z =E x]x›
6600    apply (rule "β←C"(1))
6601      apply "cqt:2[lambda]"
6602     apply (fact "cqt:2[const_var]"[axiom_inst])
6603    using "ord=Eequiv:1"[THEN "→E", OF 0[THEN "&E"(1)]].
6604  ultimately AOT_have ‹[λz z =E y]x› using "rule=E" by fast
6605  AOT_hence ‹x =E y› using "β→C"(1) by blast
6606  AOT_hence ‹x = y› by (metis "=E-simple:2" "vdash-properties:6")
6607  AOT_thus ‹x = y & ¬(x = y)› using 1 "&I" by blast
6608next
6609  AOT_assume ‹[λz z =E x] ≠ [λz z =E y]›
6610  AOT_hence 0: ‹¬([λz z =E x] = [λz z =E y])› using "≡dfE"[OF "=-infix"] by blast
6611  AOT_have ‹[λz z =E x]↓› by "cqt:2[lambda]"
6612  AOT_hence ‹[λz z =E x] = [λz z =E x]›
6613    by (metis "rule=I:1")
6614  moreover AOT_assume ‹x = y›
6615  ultimately AOT_have ‹[λz z =E x] = [λz z =E y]›
6616    using "rule=E" by fast
6617  AOT_thus ‹[λz z =E x] = [λz z =E y] & ¬([λz z =E x] = [λz z =E y])›
6618    using 0 "&I" by blast
6619qed
6620
6621AOT_theorem "ord=E2:2": ‹(O!x & O!y) → (x ≠ y ≡ [λz z = x] ≠ [λz z = y])›
6622proof (rule "→I"; rule "≡I"; rule "→I"; rule "≡dfI"[OF "=-infix"]; rule "raa-cor:2")
6623  AOT_assume 0: ‹O!x & O!y›
6624  AOT_assume ‹x ≠ y›
6625  AOT_hence 1: ‹¬(x = y)› using "≡dfE"[OF "=-infix"] by blast
6626  AOT_assume ‹[λz z = x] = [λz z = y]›
6627  moreover AOT_have ‹[λz z = x]x›
6628    apply (rule "β←C"(1))
6629    apply (fact "ord-=E=:2"[THEN "→E", OF 0[THEN "&E"(1)]])
6630     apply (fact "cqt:2[const_var]"[axiom_inst])
6631    by (simp add: "id-eq:1")
6632  ultimately AOT_have ‹[λz z = y]x› using "rule=E" by fast
6633  AOT_hence ‹x = y› using "β→C"(1) by blast
6634  AOT_thus ‹x = y & ¬(x = y)› using 1 "&I" by blast
6635next
6636  AOT_assume 0: ‹O!x & O!y›
6637  AOT_assume ‹[λz z = x] ≠ [λz z = y]›
6638  AOT_hence 1: ‹¬([λz z = x] = [λz z = y])› using "≡dfE"[OF "=-infix"] by blast
6639  AOT_have ‹[λz z = x]↓› by (fact "ord-=E=:2"[THEN "→E", OF 0[THEN "&E"(1)]])
6640  AOT_hence ‹[λz z = x] = [λz z = x]›
6641    by (metis "rule=I:1")
6642  moreover AOT_assume ‹x = y›
6643  ultimately AOT_have ‹[λz z = x] = [λz z = y]›
6644    using "rule=E" by fast
6645  AOT_thus ‹[λz z = x] = [λz z = y] & ¬([λz z = x] = [λz z = y])›
6646    using 1 "&I" by blast
6647qed
6648
6649AOT_theorem ordnecfail: ‹O!x → □¬∃F x[F]›
6650  by (meson "RM:1" "deduction-theorem" nocoder "oa-facts:1" "vdash-properties:10" "vdash-properties:1[2]")
6651
6652AOT_theorem "ab-obey:1": ‹(A!x & A!y) → (∀F (x[F] ≡ y[F]) → x = y)›
6653proof (rule "→I"; rule "→I")
6654  AOT_assume 1: ‹A!x & A!y›
6655  AOT_assume ‹∀F (x[F] ≡ y[F])›
6656  AOT_hence ‹x[F] ≡ y[F]› for F using "∀E" by blast
6657  AOT_hence ‹□(x[F] ≡ y[F])› for F by (metis "en-eq:6[1]" "≡E"(1))
6658  AOT_hence ‹∀F □(x[F] ≡ y[F])› by (rule GEN)
6659  AOT_hence ‹□∀F (x[F] ≡ y[F])› by (rule BF[THEN "→E"])
6660  AOT_thus ‹x = y›
6661    using "≡dfI"[OF "identity:1", OF "∨I"(2)] 1 "&I" by blast
6662qed
6663
6664AOT_theorem "ab-obey:2": ‹(∃F (x[F] & ¬y[F]) ∨ ∃F (y[F] & ¬x[F])) → x ≠ y›
6665proof (rule "→I"; rule "≡dfI"[OF "=-infix"]; rule "raa-cor:2")
6666  AOT_assume 1: ‹x = y›
6667  AOT_assume ‹∃F (x[F] & ¬y[F]) ∨ ∃F (y[F] & ¬x[F])›
6668  moreover {
6669    AOT_assume ‹∃F (x[F] & ¬y[F])›
6670    then AOT_obtain F where ‹x[F] & ¬y[F]› using "∃E"[rotated] by blast
6671    moreover AOT_have ‹y[F]› using calculation[THEN "&E"(1)] 1 "rule=E" by fast
6672    ultimately AOT_have ‹p & ¬p› for p by (metis "Conjunction Simplification"(2) "modus-tollens:2" "raa-cor:3")
6673  }
6674  moreover {
6675    AOT_assume ‹∃F (y[F] & ¬x[F])›
6676    then AOT_obtain F where ‹y[F] & ¬x[F]› using "∃E"[rotated] by blast
6677    moreover AOT_have ‹¬y[F]› using calculation[THEN "&E"(2)] 1 "rule=E" by fast
6678    ultimately AOT_have ‹p & ¬p› for p by (metis "Conjunction Simplification"(1) "modus-tollens:1" "raa-cor:3")
6679  }
6680  ultimately AOT_show ‹p & ¬p› for p by (metis "∨E"(3) "raa-cor:1")
6681qed
6682
6683AOT_theorem "encoders-are-abstract": ‹∃F x[F] → A!x›
6684  by (meson "deduction-theorem" "≡E"(2) "modus-tollens:2" nocoder
6685            "oa-contingent:3" "vdash-properties:1[2]")
6686
6687AOT_theorem "denote=:1": ‹∀H∃x x[H]›
6688  by (rule GEN; rule "existence:2[1]"[THEN "≡dfE"]; fact "cqt:2[const_var]"[axiom_inst])
6689
6690AOT_theorem "denote=:2": ‹∀G∃x1...∃xn x1...xn[H]›
6691  by (rule GEN; rule "existence:2"[THEN "≡dfE"]; fact "cqt:2[const_var]"[axiom_inst])
6692
6693AOT_theorem "denote=:2[2]": ‹∀G∃x1∃x2 x1x2[H]›
6694  by (rule GEN; rule "existence:2[2]"[THEN "≡dfE"]; fact "cqt:2[const_var]"[axiom_inst])
6695
6696AOT_theorem "denote=:2[3]": ‹∀G∃x1∃x2∃x3 x1x2x3[H]›
6697  by (rule GEN; rule "existence:2[3]"[THEN "≡dfE"]; fact "cqt:2[const_var]"[axiom_inst])
6698
6699AOT_theorem "denote=:2[4]": ‹∀G∃x1∃x2∃x3∃x4 x1x2x3x4[H]›
6700  by (rule GEN; rule "existence:2[4]"[THEN "≡dfE"]; fact "cqt:2[const_var]"[axiom_inst])
6701
6702AOT_theorem "denote=:3": ‹∃x x[Π] ≡ ∃H (H = Π)›
6703  using "existence:2[1]" "free-thms:1" "≡E"(2) "≡E"(5) "Commutativity of ≡" "≡Df" by blast
6704
6705AOT_theorem "denote=:4": ‹(∃x1...∃xn x1...xn[Π]) ≡ ∃H (H = Π)›
6706  using "existence:2" "free-thms:1" "≡E"(6) "≡Df" by blast
6707
6708AOT_theorem "denote=:4[2]": ‹(∃x1∃x2 x1x2[Π]) ≡ ∃H (H = Π)›
6709  using "existence:2[2]" "free-thms:1" "≡E"(6) "≡Df" by blast
6710
6711AOT_theorem "denote=:4[3]": ‹(∃x1∃x2∃x3 x1x2x3[Π]) ≡ ∃H (H = Π)›
6712  using "existence:2[3]" "free-thms:1" "≡E"(6) "≡Df" by blast
6713
6714AOT_theorem "denote=:4[4]": ‹(∃x1∃x2∃x3∃x4 x1x2x3x4[Π]) ≡ ∃H (H = Π)›
6715  using "existence:2[4]" "free-thms:1" "≡E"(6) "≡Df" by blast
6716
6717AOT_theorem "A-objects!": ‹∃!x (A!x & ∀F (x[F] ≡ φ{F}))›
6718proof (rule "uniqueness:1"[THEN "≡dfI"])
6719  AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ φ{F})›
6720    using "A-objects"[axiom_inst] "∃E"[rotated] by blast
6721  AOT_have ‹(A!β & ∀F (β[F] ≡ φ{F})) → β = a› for β
6722  proof (rule "→I")
6723    AOT_assume β_prop: ‹[A!]β & ∀F (β[F] ≡ φ{F})›
6724    AOT_hence ‹β[F] ≡ φ{F}› for F using "∀E" "&E" by blast
6725    AOT_hence ‹β[F] ≡ a[F]› for F
6726      using a_prop[THEN "&E"(2)] "∀E" "≡E"(2) "≡E"(5) "Commutativity of ≡" by fast
6727    AOT_hence ‹∀F (β[F] ≡ a[F])› by (rule GEN)
6728    AOT_thus ‹β = a›
6729      using "ab-obey:1"[THEN "→E", OF "&I"[OF β_prop[THEN "&E"(1)], OF a_prop[THEN "&E"(1)]], THEN "→E"] by blast
6730  qed
6731  AOT_hence ‹∀β ((A!β & ∀F (β[F] ≡ φ{F})) → β = a)› by (rule GEN)
6732  AOT_thus ‹∃α ([A!]α & ∀F (α[F] ≡ φ{F}) & ∀β ([A!]β & ∀F (β[F] ≡ φ{F}) → β = α)) ›
6733    using "∃I" using a_prop "&I" by fast
6734qed
6735
6736AOT_theorem "obj-oth:1": ‹∃!x (A!x & ∀F (x[F] ≡ [F]y))›
6737  using "A-objects!" by fast
6738
6739AOT_theorem "obj-oth:2": ‹∃!x (A!x & ∀F (x[F] ≡ [F]y & [F]z))›
6740  using "A-objects!" by fast
6741
6742AOT_theorem "obj-oth:3": ‹∃!x (A!x & ∀F (x[F] ≡ [F]y ∨ [F]z))›
6743  using "A-objects!" by fast
6744
6745AOT_theorem "obj-oth:4": ‹∃!x (A!x & ∀F (x[F] ≡ □[F]y))›
6746  using "A-objects!" by fast
6747
6748AOT_theorem "obj-oth:5": ‹∃!x (A!x & ∀F (x[F] ≡ F = G))›
6749  using "A-objects!" by fast
6750
6751AOT_theorem "obj-oth:6": ‹∃!x (A!x & ∀F (x[F] ≡ □∀y([G]y → [F]y)))›
6752  using "A-objects!" by fast
6753
6754AOT_theorem "A-descriptions": ‹ιx (A!x & ∀F (x[F] ≡ φ{F}))↓›
6755  by (rule "A-Exists:2"[THEN "≡E"(2)]; rule "RA[2]"; rule "A-objects!")
6756
6757AOT_act_theorem "thm-can-terms2": ‹y = ιx(A!x & ∀F (x[F] ≡ φ{F})) → (A!y & ∀F (y[F] ≡ φ{F}))›
6758  using "y-in:2" by blast
6759
6760AOT_theorem "can-ab2": ‹y = ιx(A!x & ∀F (x[F] ≡ φ{F})) →  A!y›
6761proof(rule "→I")
6762  AOT_assume ‹y = ιx(A!x & ∀F (x[F] ≡ φ{F}))›
6763  AOT_hence ‹𝒜(A!y & ∀F (y[F] ≡ φ{F}))›
6764    using "actual-desc:2"[THEN "→E"] by blast
6765  AOT_hence ‹𝒜A!y› by (metis "Act-Basic:2" "&E"(1) "≡E"(1))
6766  AOT_thus ‹A!y› by (metis "≡E"(2) "oa-facts:8")
6767qed
6768
6769AOT_act_theorem "desc-encode:1": ‹ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F}›
6770proof -
6771  AOT_have ‹ιx(A!x & ∀F (x[F] ≡ φ{F}))↓›
6772    by (simp add: "A-descriptions")
6773  AOT_hence ‹A!ιx(A!x & ∀F (x[F] ≡ φ{F})) & ∀F (ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F})›
6774    using "y-in:3"[THEN "→E"] by blast
6775  AOT_thus ‹ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F}›
6776    using "&E" "∀E" by blast
6777qed
6778
6779AOT_act_theorem "desc-encode:2": ‹ιx(A!x & ∀F (x[F] ≡ φ{F}))[G] ≡ φ{G}›
6780  using "desc-encode:1".
6781
6782AOT_theorem "desc-nec-encode:1": ‹ιx (A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ 𝒜φ{F}›
6783proof -
6784  AOT_have 0: ‹ιx(A!x & ∀F (x[F] ≡ φ{F}))↓›
6785    by (simp add: "A-descriptions")
6786  AOT_hence ‹𝒜(A!ιx(A!x & ∀F (x[F] ≡ φ{F})) & ∀F (ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F}))›
6787    using "actual-desc:4"[THEN "→E"] by blast
6788  AOT_hence ‹𝒜∀F (ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F})›
6789    using "Act-Basic:2" "&E"(2) "≡E"(1) by blast
6790  AOT_hence ‹∀F 𝒜(ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F})›
6791    using "≡E"(1) "logic-actual-nec:3" "vdash-properties:1[2]" by blast
6792  AOT_hence ‹𝒜(ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F})›
6793    using "∀E" by blast
6794  AOT_hence ‹𝒜ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ 𝒜φ{F}›
6795    using "Act-Basic:5" "≡E"(1) by blast
6796  AOT_thus ‹ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ 𝒜φ{F}›
6797    using "en-eq:10[1]"[unvarify x1, OF 0] "≡E"(6) by blast
6798qed
6799
6800AOT_theorem "desc-nec-encode:2": ‹ιx (A!x & ∀F (x[F] ≡ φ{F}))[G] ≡ 𝒜φ{G}›
6801  using "desc-nec-encode:1".
6802
6803AOT_theorem "Box-desc-encode:1": ‹□φ{G} → ιx(A!x & ∀F (x[F] ≡ φ{G}))[G]›
6804  by (rule "→I"; rule "desc-nec-encode:2"[THEN "≡E"(2)])
6805     (meson "nec-imp-act" "vdash-properties:10")
6806
6807AOT_theorem "Box-desc-encode:2": ‹□φ{G} → □(ιx(A!x & ∀F (x[F] ≡ φ{G}))[G] ≡ φ{G})›
6808proof(rule CP)
6809  AOT_assume ‹□φ{G}›
6810  AOT_hence ‹□□φ{G}› by (metis "S5Basic:6" "≡E"(1))
6811  moreover AOT_have ‹□□φ{G} → □(ιx(A!x & ∀F (x[F] ≡ φ{G}))[G] ≡ φ{G})›
6812  proof (rule RM; rule "→I")
6813    AOT_modally_strict {
6814      AOT_assume 1: ‹□φ{G}›
6815      AOT_hence ‹ιx(A!x & ∀F (x[F] ≡ φ{G}))[G]› using "Box-desc-encode:1" "→E" by blast
6816      moreover AOT_have ‹φ{G}› using 1 by (meson "qml:2" "vdash-properties:10" "vdash-properties:1[2]")
6817      ultimately AOT_show ‹ιx(A!x & ∀F (x[F] ≡ φ{G}))[G] ≡ φ{G}›
6818        using "deduction-theorem" "≡I" by simp
6819    }
6820  qed
6821  ultimately AOT_show ‹□(ιx(A!x & ∀F (x[F] ≡ φ{G}))[G] ≡ φ{G})› using "→E" by blast
6822qed
6823
6824definition rigid_condition where ‹rigid_condition φ ≡ ∀v . [v ⊨ ∀α (φ{α} → □φ{α})]›
6825syntax rigid_condition :: ‹id_position ⇒ AOT_prop› ("RIGID'_CONDITION'(_')")
6826
6827AOT_theorem "strict-can:1[E]": assumes ‹RIGID_CONDITION(φ)›
6828  shows ‹∀α (φ{α} → □φ{α})›
6829  using assms[unfolded rigid_condition_def] by auto
6830
6831AOT_theorem "strict-can:1[I]":
6832  assumes ‹⊢□ ∀α (φ{α} → □φ{α})›
6833  shows ‹RIGID_CONDITION(φ)›
6834  using assms rigid_condition_def by auto
6835
6836AOT_theorem "box-phi-a:1": assumes ‹RIGID_CONDITION(φ)›
6837  shows ‹(A!x  & ∀F (x[F] ≡ φ{F})) → □(A!x & ∀F (x[F] ≡ φ{F}))›
6838proof (rule "→I")
6839  AOT_assume a: ‹A!x & ∀F (x[F] ≡ φ{F})›
6840  AOT_hence b: ‹□A!x› by (metis "Conjunction Simplification"(1) "oa-facts:2" "vdash-properties:10")
6841  AOT_have ‹x[F] ≡ φ{F}› for F using a[THEN "&E"(2)] "∀E" by blast
6842  moreover AOT_have ‹□(x[F] → □x[F])› for F by (meson "pre-en-eq:1[1]" RN)
6843  moreover AOT_have ‹□(φ{F} → □φ{F})› for F using RN "strict-can:1[E]"[OF assms] "∀E" by blast
6844  ultimately AOT_have ‹□(x[F] ≡ φ{F})› for F
6845    using "sc-eq-box-box:5" "qml:2"[axiom_inst, THEN "→E"] "→E" "&I" by metis
6846  AOT_hence ‹∀F □(x[F] ≡ φ{F})› by (rule GEN)
6847  AOT_hence ‹□∀F (x[F] ≡ φ{F})› by (rule BF[THEN "→E"])
6848  AOT_thus ‹□([A!]x & ∀F (x[F] ≡ φ{F}))›
6849    using b "KBasic:3" "≡S"(1) "≡E"(2) by blast
6850qed
6851
6852AOT_theorem "box-phi-a:2": assumes ‹RIGID_CONDITION(φ)›
6853  shows ‹y = ιx(A!x & ∀F (x[F] ≡ φ{F})) → (A!y & ∀F (y[F] ≡ φ{F}))›
6854proof(rule "→I")
6855  AOT_assume ‹y = ιx(A!x & ∀F (x[F] ≡ φ{F}))›
6856  AOT_hence ‹𝒜(A!y & ∀F (y[F] ≡ φ{F}))› using "actual-desc:2"[THEN "→E"] by fast
6857  AOT_hence abs: ‹𝒜A!y› and ‹𝒜∀F (y[F] ≡ φ{F})›
6858    using "Act-Basic:2" "&E" "≡E"(1) by blast+
6859  AOT_hence ‹∀F 𝒜(y[F] ≡ φ{F})› by (metis "≡E"(1) "logic-actual-nec:3" "vdash-properties:1[2]")
6860  AOT_hence ‹𝒜(y[F] ≡ φ{F})› for F using "∀E" by blast
6861  AOT_hence ‹𝒜y[F] ≡ 𝒜φ{F}› for F by (metis "Act-Basic:5" "≡E"(1)) 
6862  AOT_hence ‹y[F] ≡ φ{F}› for F
6863    using "sc-eq-fur:2"[THEN "→E", OF "strict-can:1[E]"[OF assms, THEN "∀E"(2)[where β=F], THEN RN]]
6864    by (metis "en-eq:10[1]" "≡E"(6))
6865  AOT_hence ‹∀F (y[F] ≡ φ{F})› by (rule GEN)
6866  AOT_thus ‹[A!]y & ∀F (y[F] ≡ φ{F})› using abs "&I" "≡E"(2) "oa-facts:8" by blast
6867qed
6868
6869AOT_theorem "box-phi-a:3": assumes ‹RIGID_CONDITION(φ)›
6870  shows ‹ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F}›
6871  using "desc-nec-encode:2"
6872    "sc-eq-fur:2"[THEN "→E", OF "strict-can:1[E]"[OF assms, THEN "∀E"(2)[where β=F], THEN RN]]
6873    "≡E"(5) by blast
6874
6875AOT_define Null :: ‹τ ⇒ φ› ("Null'(_')") 
6876  "df-null-uni:1": ‹Null(x) ≡df A!x & ¬∃F x[F]›
6877
6878AOT_define Universal :: ‹τ ⇒ φ› ("Universal'(_')")
6879  "df-null-uni:2": ‹Universal(x) ≡df A!x & ∀F x[F]›
6880
6881AOT_theorem "null-uni-uniq:1": ‹∃!x Null(x)›
6882proof (rule "uniqueness:1"[THEN "≡dfI"])
6883  AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ ¬(F = F))›
6884    using "A-objects"[axiom_inst] "∃E"[rotated] by fast
6885  AOT_have a_null: ‹¬a[F]› for F
6886  proof (rule "raa-cor:2")
6887    AOT_assume ‹a[F]›
6888    AOT_hence ‹¬(F = F)› using a_prop[THEN "&E"(2)] "∀E" "≡E" by blast
6889    AOT_hence ‹F = F & ¬(F = F)› by (metis "id-eq:1" "raa-cor:3")
6890    AOT_thus ‹p & ¬p› for p  by (metis "raa-cor:1")
6891  qed
6892  AOT_have ‹Null(a) & ∀β (Null(β) → β = a)›
6893  proof (rule "&I")
6894    AOT_have ‹¬∃F a[F]› using a_null by (metis "instantiation" "reductio-aa:1")
6895    AOT_thus ‹Null(a)›
6896      using "df-null-uni:1"[THEN "≡dfI"] a_prop[THEN "&E"(1)] "&I" by metis
6897  next
6898    AOT_show ‹∀β (Null(β) → β = a)›
6899    proof (rule GEN; rule "→I")
6900      fix β
6901      AOT_assume a: ‹Null(β)›
6902      AOT_hence ‹¬∃F β[F]›
6903        using "df-null-uni:1"[THEN "≡dfE"] "&E" by blast
6904      AOT_hence β_null: ‹¬β[F]› for F by (metis "existential:2[const_var]" "reductio-aa:1")
6905      AOT_have ‹∀F (β[F] ≡ a[F])›
6906        apply (rule GEN; rule "≡I"; rule CP)
6907        using "raa-cor:3" β_null a_null by blast+
6908      moreover AOT_have ‹A!β› using a "df-null-uni:1"[THEN "≡dfE"] "&E" by blast
6909      ultimately AOT_show ‹β = a›
6910        using a_prop[THEN "&E"(1)] "ab-obey:1"[THEN "→E", THEN "→E"] "&I" by blast
6911    qed
6912  qed
6913  AOT_thus ‹∃α (Null(α) & ∀β (Null(β) → β = α))› using "∃I"(2) by fast
6914qed
6915
6916AOT_theorem "null-uni-uniq:2": ‹∃!x Universal(x)›
6917proof (rule "uniqueness:1"[THEN "≡dfI"])
6918  AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ F = F)›
6919    using "A-objects"[axiom_inst] "∃E"[rotated] by fast
6920  AOT_hence aF: ‹a[F]› for F using "&E" "∀E" "≡E" "id-eq:1" by fast
6921  AOT_hence ‹Universal(a)›
6922    using "df-null-uni:2"[THEN "≡dfI"] "&I" a_prop[THEN "&E"(1)] GEN by blast
6923  moreover AOT_have ‹∀β (Universal(β) → β = a)›
6924  proof (rule GEN; rule "→I")
6925    fix β
6926    AOT_assume ‹Universal(β)›
6927    AOT_hence abs_β: ‹A!β› and ‹β[F]› for F using "df-null-uni:2"[THEN "≡dfE"] "&E" "∀E" by blast+
6928    AOT_hence ‹β[F] ≡ a[F]› for F using aF by (metis "deduction-theorem" "≡I")
6929    AOT_hence ‹∀F (β[F] ≡ a[F])› by (rule GEN)
6930    AOT_thus ‹β = a›
6931      using a_prop[THEN "&E"(1)] "ab-obey:1"[THEN "→E", THEN "→E"] "&I" abs_β by blast
6932  qed
6933  ultimately AOT_show ‹∃α (Universal(α) & ∀β (Universal(β) → β = α))›
6934    using "&I" "∃I" by fast
6935qed
6936
6937AOT_theorem "null-uni-uniq:3": ‹ιx Null(x)↓›
6938  using "A-Exists:2" "RA[2]" "≡E"(2) "null-uni-uniq:1" by blast
6939
6940AOT_theorem "null-uni-uniq:4": ‹ιx Universal(x)↓›
6941  using "A-Exists:2" "RA[2]" "≡E"(2) "null-uni-uniq:2" by blast
6942
6943AOT_define Null_object :: ‹κs› (‹a∅›)
6944  "df-null-uni-terms:1": ‹a∅ =df ιx Null(x)›
6945
6946AOT_define Universal_object :: ‹κs› (‹aV›)
6947  "df-null-uni-terms:2": ‹aV =df ιx Universal(x)›
6948
6949AOT_theorem "null-uni-facts:1": ‹Null(x) → □Null(x)›
6950proof (rule "→I")
6951  AOT_assume ‹Null(x)›
6952  AOT_hence x_abs: ‹A!x› and x_null: ‹¬∃F x[F]›
6953    using "df-null-uni:1"[THEN "≡dfE"] "&E" by blast+
6954  AOT_have ‹¬x[F]› for F using x_null
6955    using "existential:2[const_var]" "reductio-aa:1"
6956    by metis
6957  AOT_hence ‹□¬x[F]› for F by (metis "en-eq:7[1]" "≡E"(1))
6958  AOT_hence ‹∀F □¬x[F]› by (rule GEN)
6959  AOT_hence ‹□∀F ¬x[F]› by (rule BF[THEN "→E"])
6960  moreover AOT_have ‹□∀F ¬x[F] → □¬∃F x[F]›
6961    apply (rule RM)
6962    by (metis (full_types) "instantiation" "cqt:2[const_var]" "deduction-theorem"
6963                           "reductio-aa:1" "rule-ui:1" "vdash-properties:1[2]")
6964  ultimately AOT_have ‹□¬∃F x[F]›
6965    by (metis "→E")
6966  moreover AOT_have ‹□A!x› using x_abs
6967    using "oa-facts:2" "vdash-properties:10" by blast
6968  ultimately AOT_have r: ‹□(A!x & ¬∃F x[F])›
6969    by (metis "KBasic:3" "&I" "≡E"(3) "raa-cor:3")
6970  AOT_show ‹□Null(x)›
6971    by (AOT_subst "«Null(x)»" "«A!x & ¬∃F x[F]»")
6972       (auto simp: "df-null-uni:1" "≡Df" r)
6973qed  
6974
6975AOT_theorem "null-uni-facts:2": ‹Universal(x) → □Universal(x)›
6976proof (rule "→I")
6977  AOT_assume ‹Universal(x)›
6978  AOT_hence x_abs: ‹A!x› and x_univ: ‹∀F x[F]›
6979    using "df-null-uni:2"[THEN "≡dfE"] "&E" by blast+
6980  AOT_have ‹x[F]› for F using x_univ "∀E" by blast
6981  AOT_hence ‹□x[F]› for F by (metis "en-eq:2[1]" "≡E"(1))
6982  AOT_hence ‹∀F □x[F]› by (rule GEN)
6983  AOT_hence ‹□∀F x[F]› by (rule BF[THEN "→E"])
6984  moreover AOT_have ‹□A!x› using x_abs
6985    using "oa-facts:2" "vdash-properties:10" by blast
6986  ultimately AOT_have r: ‹□(A!x & ∀F x[F])›
6987    by (metis "KBasic:3" "&I" "≡E"(3) "raa-cor:3")
6988  AOT_show ‹□Universal(x)›
6989    by (AOT_subst "«Universal(x)»" "«A!x & ∀F x[F]»")
6990       (auto simp add: "df-null-uni:2" "≡Df" r)
6991qed
6992
6993AOT_theorem "null-uni-facts:3": ‹Null(a∅)›
6994  apply (rule "=dfI"(2)[OF "df-null-uni-terms:1"])
6995   apply (simp add: "null-uni-uniq:3")
6996  using "actual-desc:4"[THEN "→E", OF "null-uni-uniq:3"]
6997    "sc-eq-fur:2"[THEN "→E", OF "null-uni-facts:1"[unvarify x, THEN RN, OF "null-uni-uniq:3"], THEN "≡E"(1)]
6998  by blast
6999
7000AOT_theorem "null-uni-facts:4": ‹Universal(aV)›
7001  apply (rule "=dfI"(2)[OF "df-null-uni-terms:2"])
7002   apply (simp add: "null-uni-uniq:4")
7003  using "actual-desc:4"[THEN "→E", OF "null-uni-uniq:4"]
7004    "sc-eq-fur:2"[THEN "→E", OF "null-uni-facts:2"[unvarify x, THEN RN, OF "null-uni-uniq:4"], THEN "≡E"(1)]
7005  by blast
7006
7007AOT_theorem "null-uni-facts:5": ‹a∅ ≠ aV›
7008proof (rule "=dfI"(2)[OF "df-null-uni-terms:1", OF "null-uni-uniq:3"];
7009    rule "=dfI"(2)[OF "df-null-uni-terms:2", OF "null-uni-uniq:4"];
7010    rule "≡dfI"[OF "=-infix"];
7011    rule "raa-cor:2")
7012  AOT_obtain x where nullx: ‹Null(x)›
7013    by (metis "instantiation" "df-null-uni-terms:1" "existential:1" "null-uni-facts:3"
7014              "null-uni-uniq:3" "rule-id-df:2:b[zero]")
7015  AOT_hence act_null: ‹𝒜Null(x)› by (metis "nec-imp-act" "null-uni-facts:1" "vdash-properties:10")
7016  AOT_assume ‹ιx Null(x) = ιx Universal(x)›
7017  AOT_hence ‹𝒜∀x(Null(x) ≡ Universal(x))›
7018    using "actual-desc:5"[THEN "→E"] by blast
7019  AOT_hence ‹∀x 𝒜(Null(x) ≡ Universal(x))›
7020    by (metis "≡E"(1) "logic-actual-nec:3" "vdash-properties:1[2]")
7021  AOT_hence ‹𝒜Null(x) ≡ 𝒜Universal(x)›
7022    using "Act-Basic:5" "≡E"(1) "rule-ui:3" by blast
7023  AOT_hence ‹𝒜Universal(x)› using act_null "≡E" by blast
7024  AOT_hence ‹Universal(x)› by (metis RN "≡E"(1) "null-uni-facts:2" "sc-eq-fur:2" "vdash-properties:10")
7025  AOT_hence ‹∀F x[F]› using "≡dfE"[OF "df-null-uni:2"] "&E" by metis
7026  moreover AOT_have ‹¬∃F x[F]› using nullx "≡dfE"[OF "df-null-uni:1"] "&E" by metis
7027  ultimately AOT_show ‹p & ¬p› for p by (metis "cqt-further:1" "raa-cor:3" "vdash-properties:10")
7028qed
7029
7030AOT_theorem "null-uni-facts:6": ‹a∅ = ιx(A!x & ∀F (x[F] ≡ F ≠ F))›
7031proof (rule "ab-obey:1"[unvarify x y, THEN "→E", THEN "→E"])
7032  AOT_show ‹ιx([A!]x & ∀F (x[F] ≡ F ≠ F))↓›
7033    by (simp add: "A-descriptions")
7034next
7035  AOT_show ‹a∅↓›
7036    by (rule "=dfI"(2)[OF "df-null-uni-terms:1", OF "null-uni-uniq:3"])
7037       (simp add: "null-uni-uniq:3")
7038next
7039  AOT_have ‹ιx([A!]x & ∀F (x[F] ≡ F ≠ F))↓›
7040    by (simp add: "A-descriptions")
7041  AOT_hence 1: ‹ιx([A!]x & ∀F (x[F] ≡ F ≠ F)) = ιx([A!]x & ∀F (x[F] ≡ F ≠ F))›
7042    using "rule=I:1" by blast
7043  AOT_show ‹[A!]a∅ & [A!]ιx([A!]x & ∀F (x[F] ≡ F ≠ F))›
7044    apply (rule "=dfI"(2)[OF "df-null-uni-terms:1", OF "null-uni-uniq:3"]; rule "&I")
7045    apply (meson "≡dfE" "Conjunction Simplification"(1) "df-null-uni:1" "df-null-uni-terms:1" "null-uni-facts:3" "null-uni-uniq:3" "rule-id-df:2:a[zero]" "vdash-properties:10")
7046    using "can-ab2"[unvarify y, OF "A-descriptions", THEN "→E", OF 1].
7047next
7048  AOT_show ‹∀F (a∅[F] ≡ ιx([A!]x & ∀F (x[F] ≡ F ≠ F))[F])›
7049  proof (rule GEN)
7050    fix F
7051    AOT_have ‹¬a∅[F]›
7052      by (rule "=dfI"(2)[OF "df-null-uni-terms:1", OF "null-uni-uniq:3"])
7053         (metis (no_types, lifting) "≡dfE" "&E"(2) "∨I"(2) "∨E"(3)
7054                "df-null-uni:1" "df-null-uni-terms:1" "existential:2[const_var]" "null-uni-facts:3"
7055                "raa-cor:2" "rule-id-df:2:a[zero]" "russell-axiom[enc,1].ψ_denotes_asm")
7056    moreover AOT_have ‹¬ιx([A!]x & ∀F (x[F] ≡ F ≠ F))[F]›
7057    proof(rule "raa-cor:2")
7058      AOT_assume 0: ‹ιx([A!]x & ∀F (x[F] ≡ F ≠ F))[F]›
7059      AOT_hence ‹𝒜(F ≠ F)› using "desc-nec-encode:2"[THEN "≡E"(1), OF 0] by blast
7060      moreover AOT_have ‹¬𝒜(F ≠ F)›
7061        using "≡dfE" "id-act:2" "id-eq:1" "≡E"(2) "=-infix" "raa-cor:3" by blast
7062      ultimately AOT_show ‹𝒜(F ≠ F) & ¬𝒜(F ≠ F)› by (rule "&I")
7063    qed
7064    ultimately AOT_show ‹a∅[F] ≡ ιx([A!]x & ∀F (x[F] ≡ F ≠ F))[F]›
7065      using "deduction-theorem" "≡I" "raa-cor:4" by blast
7066  qed
7067qed
7068
7069AOT_theorem "null-uni-facts:7": ‹aV = ιx(A!x & ∀F (x[F] ≡ F = F))›
7070proof (rule "ab-obey:1"[unvarify x y, THEN "→E", THEN "→E"])
7071  AOT_show ‹ιx([A!]x & ∀F (x[F] ≡ F = F))↓›
7072    by (simp add: "A-descriptions")
7073next
7074  AOT_show ‹aV↓›
7075    by (rule "=dfI"(2)[OF "df-null-uni-terms:2", OF "null-uni-uniq:4"])
7076       (simp add: "null-uni-uniq:4")
7077next
7078  AOT_have ‹ιx([A!]x & ∀F (x[F] ≡ F = F))↓›
7079    by (simp add: "A-descriptions")
7080  AOT_hence 1: ‹ιx([A!]x & ∀F (x[F] ≡ F = F)) = ιx([A!]x & ∀F (x[F] ≡ F = F))›
7081    using "rule=I:1" by blast
7082  AOT_show ‹[A!]aV & [A!]ιx([A!]x & ∀F (x[F] ≡ F = F))›
7083    apply (rule "=dfI"(2)[OF "df-null-uni-terms:2", OF "null-uni-uniq:4"]; rule "&I")
7084    apply (meson "≡dfE" "Conjunction Simplification"(1) "df-null-uni:2" "df-null-uni-terms:2" "null-uni-facts:4" "null-uni-uniq:4" "rule-id-df:2:a[zero]" "vdash-properties:10")
7085    using "can-ab2"[unvarify y, OF "A-descriptions", THEN "→E", OF 1].
7086next
7087  AOT_show ‹∀F (aV[F] ≡ ιx([A!]x & ∀F (x[F] ≡ F = F))[F])›
7088  proof (rule GEN)
7089    fix F
7090    AOT_have ‹aV[F]›
7091      apply (rule "=dfI"(2)[OF "df-null-uni-terms:2", OF "null-uni-uniq:4"])
7092      using "≡dfE" "&E"(2) "df-null-uni:2" "df-null-uni-terms:2" "null-uni-facts:4" "null-uni-uniq:4" "rule-id-df:2:a[zero]" "rule-ui:3" by blast
7093    moreover AOT_have ‹ιx([A!]x & ∀F (x[F] ≡ F = F))[F]›
7094      using "RA[2]" "desc-nec-encode:2" "id-eq:1" "≡E"(2) by fastforce
7095    ultimately AOT_show ‹aV[F] ≡ ιx([A!]x & ∀F (x[F] ≡ F = F))[F]›
7096      using "deduction-theorem" "≡I" by simp
7097  qed
7098qed
7099
7100AOT_theorem "aclassical:1": ‹∀R∃x∃y(A!x & A!y & x ≠ y & [λz [R]zx] = [λz [R]zy])›
7101proof(rule GEN)
7102  fix R
7103  AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ ∃y(A!y & F = [λz [R]zy] & ¬y[F]))›
7104    using "A-objects"[axiom_inst] "∃E"[rotated] by fast
7105  AOT_have a_enc: ‹a[λz [R]za]›
7106  proof (rule "raa-cor:1")
7107    AOT_assume 0: ‹¬a[λz [R]za]›
7108    AOT_hence ‹¬∃y(A!y & [λz [R]za] = [λz [R]zy] & ¬y[λz [R]za])›
7109      by (rule a_prop[THEN "&E"(2), THEN "∀E"(1)[where τ="«[λz [R]za]»"],
7110                THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated])
7111         "cqt:2[lambda]"
7112    AOT_hence ‹∀y ¬(A!y & [λz [R]za] = [λz [R]zy] & ¬y[λz [R]za])›
7113      using "cqt-further:4" "vdash-properties:10" by blast
7114    AOT_hence ‹¬(A!a & [λz [R]za] = [λz [R]za] & ¬a[λz [R]za])› using "∀E" by blast
7115    AOT_hence ‹(A!a & [λz [R]za] = [λz [R]za]) → a[λz [R]za]›
7116      by (metis "&I" "deduction-theorem" "raa-cor:3")
7117    moreover AOT_have ‹[λz [R]za] = [λz [R]za]›
7118      by (rule "=I") "cqt:2[lambda]"
7119    ultimately AOT_have ‹a[λz [R]za]› using a_prop[THEN "&E"(1)] "→E" "&I" by blast
7120    AOT_thus ‹a[λz [R]za] & ¬a[λz [R]za]›
7121      using 0 "&I" by blast
7122  qed
7123  AOT_hence ‹∃y(A!y & [λz [R]za] = [λz [R]zy] & ¬y[λz [R]za])›
7124    by (rule a_prop[THEN "&E"(2), THEN "∀E"(1), THEN "≡E"(1), rotated]) "cqt:2[lambda]"
7125  then AOT_obtain b where b_prop: ‹A!b & [λz [R]za] = [λz [R]zb] & ¬b[λz [R]za]›
7126    using "∃E"[rotated] by blast
7127  AOT_have ‹a ≠ b›
7128    apply (rule "≡dfI"[OF "=-infix"])
7129    using a_enc b_prop[THEN "&E"(2)]
7130    using "¬¬I" "rule=E" id_sym "≡E"(4) "oth-class-taut:3:a" "raa-cor:3" "reductio-aa:1" by fast
7131  AOT_hence ‹A!a & A!b & a ≠ b & [λz [R]za] = [λz [R]zb]›
7132    using b_prop "&E" a_prop "&I" by meson
7133  AOT_hence ‹∃y (A!a & A!y & a ≠ y & [λz [R]za] = [λz [R]zy])› by (rule "∃I")
7134  AOT_thus ‹∃x∃y (A!x & A!y & x ≠ y & [λz [R]zx] = [λz [R]zy])› by (rule "∃I")
7135qed
7136
7137AOT_theorem "aclassical:2": ‹∀R∃x∃y(A!x & A!y & x ≠ y & [λz [R]xz] = [λz [R]yz])›
7138proof(rule GEN)
7139  fix R
7140  AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ ∃y(A!y & F = [λz [R]yz] & ¬y[F]))›
7141    using "A-objects"[axiom_inst] "∃E"[rotated] by fast
7142  AOT_have a_enc: ‹a[λz [R]az]›
7143  proof (rule "raa-cor:1")
7144    AOT_assume 0: ‹¬a[λz [R]az]›
7145    AOT_hence ‹¬∃y(A!y & [λz [R]az] = [λz [R]yz] & ¬y[λz [R]az])›
7146      by (rule a_prop[THEN "&E"(2), THEN "∀E"(1)[where τ="«[λz [R]az]»"],
7147                THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated])
7148         "cqt:2[lambda]"
7149    AOT_hence ‹∀y ¬(A!y & [λz [R]az] = [λz [R]yz] & ¬y[λz [R]az])›
7150      using "cqt-further:4" "vdash-properties:10" by blast
7151    AOT_hence ‹¬(A!a & [λz [R]az] = [λz [R]az] & ¬a[λz [R]az])› using "∀E" by blast
7152    AOT_hence ‹(A!a & [λz [R]az] = [λz [R]az]) → a[λz [R]az]›
7153      by (metis "&I" "deduction-theorem" "raa-cor:3")
7154    moreover AOT_have ‹[λz [R]az] = [λz [R]az]›
7155      by (rule "=I") "cqt:2[lambda]"
7156    ultimately AOT_have ‹a[λz [R]az]› using a_prop[THEN "&E"(1)] "→E" "&I" by blast
7157    AOT_thus ‹a[λz [R]az] & ¬a[λz [R]az]›
7158      using 0 "&I" by blast
7159  qed
7160  AOT_hence ‹∃y(A!y & [λz [R]az] = [λz [R]yz] & ¬y[λz [R]az])›
7161    by (rule a_prop[THEN "&E"(2), THEN "∀E"(1), THEN "≡E"(1), rotated]) "cqt:2[lambda]"
7162  then AOT_obtain b where b_prop: ‹A!b & [λz [R]az] = [λz [R]bz] & ¬b[λz [R]az]›
7163    using "∃E"[rotated] by blast
7164  AOT_have ‹a ≠ b›
7165    apply (rule "≡dfI"[OF "=-infix"])
7166    using a_enc b_prop[THEN "&E"(2)]
7167    using "¬¬I" "rule=E" id_sym "≡E"(4) "oth-class-taut:3:a" "raa-cor:3" "reductio-aa:1" by fast
7168  AOT_hence ‹A!a & A!b & a ≠ b & [λz [R]az] = [λz [R]bz]›
7169    using b_prop "&E" a_prop "&I" by meson
7170  AOT_hence ‹∃y (A!a & A!y & a ≠ y & [λz [R]az] = [λz [R]yz])› by (rule "∃I")
7171  AOT_thus ‹∃x∃y (A!x & A!y & x ≠ y & [λz [R]xz] = [λz [R]yz])› by (rule "∃I")
7172qed
7173
7174AOT_theorem "aclassical:3": ‹∀F∃x∃y(A!x & A!y & x ≠ y & [λ [F]x] = [λ [F]y])›
7175proof(rule GEN)
7176  fix R
7177  AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ ∃y(A!y & F = [λz [R]y] & ¬y[F]))›
7178    using "A-objects"[axiom_inst] "∃E"[rotated] by fast
7179  AOT_have ‹[λz [R]a]↓› by "cqt:2[lambda]"
7180  (* TODO: S should no longer be necessary *)
7181  then AOT_obtain S where S_def: ‹S = [λz [R]a]›
7182    by (metis "instantiation" "rule=I:1" "existential:1" id_sym)
7183  AOT_have a_enc: ‹a[S]›
7184  proof (rule "raa-cor:1")
7185    AOT_assume 0: ‹¬a[S]›
7186    AOT_hence ‹¬∃y(A!y & S = [λz [R]y] & ¬y[S])›
7187      by (rule a_prop[THEN "&E"(2), THEN "∀E"(2)[where β=S],
7188                THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated]) 
7189    AOT_hence ‹∀y ¬(A!y & S = [λz [R]y] & ¬y[S])›
7190      using "cqt-further:4" "vdash-properties:10" by blast
7191    AOT_hence ‹¬(A!a & S = [λz [R]a] & ¬a[S])› using "∀E" by blast
7192    AOT_hence ‹(A!a & S = [λz [R]a]) → a[S]›
7193      by (metis "&I" "deduction-theorem" "raa-cor:3")
7194    moreover AOT_have ‹S = [λz [R]a]› using S_def .
7195    ultimately AOT_have ‹a[S]› using a_prop[THEN "&E"(1)] "→E" "&I" by blast
7196    AOT_thus ‹a[λz [R]a] & ¬a[λz [R]a]›  by (metis "0" "raa-cor:3") 
7197  qed
7198  AOT_hence ‹∃y(A!y & S = [λz [R]y] & ¬y[S])›
7199    by (rule a_prop[THEN "&E"(2), THEN "∀E"(2), THEN "≡E"(1), rotated])
7200  then AOT_obtain b where b_prop: ‹A!b & S = [λz [R]b] & ¬b[S]›
7201    using "∃E"[rotated] by blast
7202  AOT_have 1: ‹a ≠ b›
7203    apply (rule "≡dfI"[OF "=-infix"])
7204    using a_enc b_prop[THEN "&E"(2)]
7205    using "¬¬I" "rule=E" id_sym "≡E"(4) "oth-class-taut:3:a" "raa-cor:3" "reductio-aa:1" by fast
7206  AOT_have a: ‹[λ [R]a] = ([R]a)›
7207    apply (rule "lambda-predicates:3[zero]"[axiom_inst, unvarify p])
7208    by (meson "log-prop-prop:2")
7209  AOT_have b: ‹[λ [R]b] = ([R]b)›
7210    apply (rule "lambda-predicates:3[zero]"[axiom_inst, unvarify p])
7211    by (meson "log-prop-prop:2")
7212  AOT_have ‹[λ [R]a] = [λ [R]b]›
7213    apply (rule "rule=E"[rotated, OF a[THEN id_sym]])
7214    apply (rule "rule=E"[rotated, OF b[THEN id_sym]])
7215    apply (rule "identity:4"[THEN "≡dfI", OF "&I", rotated])
7216     apply (rule "rule=E"[rotated, OF S_def])
7217    using b_prop "&E" apply blast
7218    apply (safe intro!: "&I")
7219    by (simp add: "log-prop-prop:2")+
7220  AOT_hence ‹A!a & A!b & a ≠ b & [λ [R]a] = [λ [R]b]›
7221    using 1 a_prop[THEN "&E"(1)] b_prop[THEN "&E"(1), THEN "&E"(1)] "&I" by auto
7222  AOT_hence ‹∃y (A!a & A!y & a ≠ y & [λ [R]a] = [λ [R]y])› by (rule "∃I")
7223  AOT_thus ‹∃x∃y (A!x & A!y & x ≠ y & [λ [R]x] = [λ [R]y])› by (rule "∃I")
7224qed
7225
7226AOT_theorem aclassical2: ‹∃x∃y (A!x & A!y & x ≠ y & ∀F ([F]x ≡ [F]y))›
7227proof -
7228  AOT_have ‹∃x ∃y ([A!]x & [A!]y & x ≠ y &
7229               [λz [λxy ∀F ([F]x ≡ [F]y)]zx] = [λz [λxy ∀F ([F]x ≡ [F]y)]zy])›
7230    by (rule "aclassical:1"[THEN "∀E"(1)[where τ="«[λxy ∀F ([F]x ≡ [F]y)]»"]])
7231       "cqt:2[lambda]"
7232  then AOT_obtain x where ‹∃y ([A!]x & [A!]y & x ≠ y &
7233               [λz [λxy ∀F ([F]x ≡ [F]y)]zx] = [λz [λxy ∀F ([F]x ≡ [F]y)]zy])›
7234    using "∃E"[rotated] by blast
7235  then AOT_obtain y where 0: ‹([A!]x & [A!]y & x ≠ y &
7236               [λz [λxy ∀F ([F]x ≡ [F]y)]zx] = [λz [λxy ∀F ([F]x ≡ [F]y)]zy])›
7237    using "∃E"[rotated] by blast
7238  AOT_have ‹[λz [λxy ∀F ([F]x ≡ [F]y)]zx]x›
7239    apply (rule "β←C"(1))
7240      apply "cqt:2[lambda]"
7241     apply (fact "cqt:2[const_var]"[axiom_inst])
7242    apply (rule "β←C"(1))
7243      apply "cqt:2[lambda]"
7244    apply (simp add: "&I" "ex:1:a" prod_denotesI "rule-ui:3")
7245    by (simp add: "oth-class-taut:3:a" "universal-cor")
7246  AOT_hence ‹[λz [λxy ∀F ([F]x ≡ [F]y)]zy]x›
7247    by (rule "rule=E"[rotated, OF 0[THEN "&E"(2)]])
7248  AOT_hence ‹[λxy ∀F ([F]x ≡ [F]y)]xy›
7249    by (rule "β→C"(1))
7250  AOT_hence ‹∀F ([F]x ≡ [F]y)›
7251    using "β→C"(1) old.prod.case by fast
7252  AOT_hence ‹[A!]x & [A!]y & x ≠ y & ∀F ([F]x ≡ [F]y)› using 0 "&E" "&I" by blast
7253  AOT_hence ‹∃y ([A!]x & [A!]y & x ≠ y & ∀F ([F]x ≡ [F]y))› by (rule "∃I")
7254  AOT_thus ‹∃x∃y ([A!]x & [A!]y & x ≠ y & ∀F ([F]x ≡ [F]y))› by (rule "∃I"(2))
7255qed
7256
7257AOT_theorem "kirchner-thm:1": ‹[λx φ{x}]↓ ≡ □∀x∀y(∀F([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7258proof(rule "≡I"; rule "→I")
7259  AOT_assume ‹[λx φ{x}]↓›
7260  AOT_hence ‹□[λx φ{x}]↓› by (metis "exist-nec" "vdash-properties:10")
7261  moreover AOT_have ‹□[λx φ{x}]↓ → □∀x∀y(∀F([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7262  proof (rule "RM:1"; rule "→I"; rule GEN; rule GEN; rule "→I")
7263    AOT_modally_strict {
7264      fix x y
7265      AOT_assume 0: ‹[λx φ{x}]↓›
7266      moreover AOT_assume ‹∀F([F]x ≡ [F]y)›
7267      ultimately AOT_have ‹[λx φ{x}]x ≡ [λx φ{x}]y›
7268        using "∀E" by blast
7269      AOT_thus ‹(φ{x} ≡ φ{y})›
7270        using "beta-C-meta"[THEN "→E", OF 0] "≡E"(6) by meson
7271    }
7272  qed
7273  ultimately AOT_show ‹□∀x∀y(∀F([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7274    using "→E" by blast
7275next
7276  AOT_have ‹□∀x∀y(∀F([F]x ≡ [F]y) → (φ{x} ≡ φ{y})) → □∀y(∃x(∀F([F]x ≡ [F]y) & φ{x}) ≡ φ{y})›
7277  proof(rule "RM:1"; rule "→I"; rule GEN)
7278    AOT_modally_strict {
7279      AOT_assume ‹∀x∀y(∀F([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7280      AOT_hence indisc: ‹φ{x} ≡ φ{y}› if ‹∀F([F]x ≡ [F]y)› for x y
7281        using "∀E"(2) "→E" that by blast
7282      AOT_show ‹(∃x(∀F([F]x ≡ [F]y) & φ{x}) ≡ φ{y})› for y
7283      proof (rule "raa-cor:1")
7284        AOT_assume ‹¬(∃x(∀F([F]x ≡ [F]y) & φ{x}) ≡ φ{y})›
7285        AOT_hence ‹(∃x(∀F([F]x ≡ [F]y) & φ{x}) & ¬φ{y}) ∨ (¬(∃x(∀F([F]x ≡ [F]y) & φ{x})) & φ{y})›
7286          using "≡E"(1) "oth-class-taut:4:h" by blast
7287        moreover {
7288          AOT_assume 0: ‹∃x(∀F([F]x ≡ [F]y) & φ{x}) & ¬φ{y}›
7289          AOT_obtain a where ‹∀F([F]a ≡ [F]y) & φ{a}›
7290            using "∃E"[rotated, OF 0[THEN "&E"(1)]]  by blast
7291          AOT_hence ‹φ{y}› using indisc[THEN "≡E"(1)] "&E" by blast
7292          AOT_hence ‹p & ¬p› for p using 0[THEN "&E"(2)] "&I" "raa-cor:3" by blast
7293        }
7294        moreover {
7295          AOT_assume 0: ‹(¬(∃x(∀F([F]x ≡ [F]y) & φ{x})) & φ{y})›
7296          AOT_hence ‹∀x ¬(∀F([F]x ≡ [F]y) & φ{x})›
7297            using "&E"(1) "cqt-further:4" "→E" by blast
7298          AOT_hence ‹¬(∀F([F]y ≡ [F]y) & φ{y})› using "∀E" by blast
7299          AOT_hence ‹¬∀F([F]y ≡ [F]y) ∨ ¬φ{y}›
7300            using "≡E"(1) "oth-class-taut:5:c" by blast
7301          moreover AOT_have ‹∀F([F]y ≡ [F]y)› by (simp add: "oth-class-taut:3:a" "universal-cor")
7302          ultimately AOT_have ‹¬φ{y}› by (metis "¬¬I" "∨E"(2))
7303          AOT_hence ‹p & ¬p› for p using 0[THEN "&E"(2)] "&I" "raa-cor:3" by blast
7304        }
7305        ultimately AOT_show ‹p & ¬p› for p using "∨E"(3) "raa-cor:1" by blast
7306      qed
7307    }
7308  qed
7309  moreover AOT_assume ‹□∀x∀y(∀F([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7310  ultimately AOT_have ‹□∀y(∃x(∀F([F]x ≡ [F]y) & φ{x}) ≡ φ{y})›
7311    using "→E" by blast
7312  AOT_thus ‹[λx φ{x}]↓›
7313    by (rule "safe-ext"[axiom_inst, THEN "→E", OF "&I", rotated]) "cqt:2[lambda]"
7314qed
7315
7316AOT_theorem "kirchner-thm:2": ‹[λx1...xn φ{x1...xn}]↓ ≡ □∀x1...∀xn∀y1...∀yn(∀F([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7317proof(rule "≡I"; rule "→I")
7318  AOT_assume ‹[λx1...xn φ{x1...xn}]↓›
7319  AOT_hence ‹□[λx1...xn φ{x1...xn}]↓› by (metis "exist-nec" "vdash-properties:10")
7320  moreover AOT_have ‹□[λx1...xn φ{x1...xn}]↓ → □∀x1...∀xn∀y1...∀yn(∀F([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7321  proof (rule "RM:1"; rule "→I"; rule GEN; rule GEN; rule "→I")
7322    AOT_modally_strict {
7323      fix x1xn y1yn :: ‹'a AOT_var›
7324      AOT_assume 0: ‹[λx1...xn φ{x1...xn}]↓›
7325      moreover AOT_assume ‹∀F([F]x1...xn ≡ [F]y1...yn)›
7326      ultimately AOT_have ‹[λx1...xn φ{x1...xn}]x1...xn ≡ [λx1...xn φ{x1...xn}]y1...yn›
7327        using "∀E" by blast
7328      AOT_thus ‹(φ{x1...xn} ≡ φ{y1...yn})›
7329        using "beta-C-meta"[THEN "→E", OF 0] "≡E"(6) by meson
7330    }
7331  qed
7332  ultimately AOT_show ‹□∀x1...∀xn∀y1...∀yn(∀F([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7333    using "→E" by blast
7334next
7335  AOT_have ‹□(∀x1...∀xn∀y1...∀yn(∀F([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))) →
7336            □∀y1...∀yn((∃x1...∃xn(∀F([F]x1...xn ≡ [F]y1...yn) & φ{x1...xn})) ≡ φ{y1...yn})›
7337  proof(rule "RM:1"; rule "→I"; rule GEN)
7338    AOT_modally_strict {
7339      AOT_assume ‹∀x1...∀xn∀y1...∀yn(∀F([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7340      AOT_hence indisc: ‹φ{x1...xn} ≡ φ{y1...yn}› if ‹∀F([F]x1...xn ≡ [F]y1...yn)› for x1xn y1yn
7341        using "∀E"(2) "→E" that by blast
7342      AOT_show ‹(∃x1...∃xn(∀F([F]x1...xn ≡ [F]y1...yn) & φ{x1...xn})) ≡ φ{y1...yn}› for y1yn
7343      proof (rule "raa-cor:1")
7344        AOT_assume ‹¬((∃x1...∃xn(∀F([F]x1...xn ≡ [F]y1...yn) & φ{x1...xn})) ≡ φ{y1...yn})›
7345        AOT_hence ‹((∃x1...∃xn(∀F([F]x1...xn ≡ [F]y1...yn) & φ{x1...xn})) & ¬φ{y1...yn}) ∨
7346                    (¬(∃x1...∃xn(∀F([F]x1...xn ≡ [F]y1...yn) & φ{x1...xn})) & φ{y1...yn})›
7347          using "≡E"(1) "oth-class-taut:4:h" by blast
7348        moreover {
7349          AOT_assume 0: ‹(∃x1...∃xn(∀F([F]x1...xn ≡ [F]y1...yn) & φ{x1...xn})) & ¬φ{y1...yn}›
7350          AOT_obtain a1an where ‹∀F([F]a1...an ≡ [F]y1...yn) & φ{a1...an}›
7351            using "∃E"[rotated, OF 0[THEN "&E"(1)]]  by blast
7352          AOT_hence ‹φ{y1...yn}› using indisc[THEN "≡E"(1)] "&E" by blast
7353          AOT_hence ‹p & ¬p› for p using 0[THEN "&E"(2)] "&I" "raa-cor:3" by blast
7354        }
7355        moreover {
7356          AOT_assume 0: ‹(¬((∃x1...∃xn(∀F([F]x1...xn ≡ [F]y1...yn) & φ{x1...xn}))) & φ{y1...yn})›
7357          AOT_hence ‹∀x1...∀xn ¬(∀F([F]x1...xn ≡ [F]y1...yn) & φ{x1...xn})›
7358            using "&E"(1) "cqt-further:4" "→E" by blast
7359          AOT_hence ‹¬(∀F([F]y1...yn ≡ [F]y1...yn) & φ{y1...yn})› using "∀E" by blast
7360          AOT_hence ‹¬∀F([F]y1...yn ≡ [F]y1...yn) ∨ ¬φ{y1...yn}›
7361            using "≡E"(1) "oth-class-taut:5:c" by blast
7362          moreover AOT_have ‹∀F([F]y1...yn ≡ [F]y1...yn)›
7363            by (simp add: "oth-class-taut:3:a" "universal-cor")
7364          ultimately AOT_have ‹¬φ{y1...yn}› by (metis "¬¬I" "∨E"(2))
7365          AOT_hence ‹p & ¬p› for p using 0[THEN "&E"(2)] "&I" "raa-cor:3" by blast
7366        }
7367        ultimately AOT_show ‹p & ¬p› for p using "∨E"(3) "raa-cor:1" by blast
7368      qed
7369    }
7370  qed
7371  moreover AOT_assume ‹□∀x1...∀xn∀y1...∀yn(∀F([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7372  ultimately AOT_have ‹□∀y1...∀yn((∃x1...∃xn(∀F([F]x1...xn ≡ [F]y1...yn) & φ{x1...xn})) ≡ φ{y1...yn})›
7373    using "→E" by blast
7374  AOT_thus ‹[λx1...xn φ{x1...xn}]↓›
7375    by (rule "safe-ext"[axiom_inst, THEN "→E", OF "&I", rotated]) "cqt:2[lambda]"
7376qed
7377
7378AOT_theorem "kirchner-thm-cor:1": ‹[λx φ{x}]↓ → ∀x∀y(∀F([F]x ≡ [F]y) → □(φ{x} ≡ φ{y}))›
7379proof(rule "→I"; rule GEN; rule GEN; rule "→I")
7380  fix x y
7381  AOT_assume ‹[λx φ{x}]↓›
7382  AOT_hence ‹□∀x∀y (∀F ([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7383    by (rule "kirchner-thm:1"[THEN "≡E"(1)])
7384  AOT_hence ‹∀x□∀y (∀F ([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7385    using CBF[THEN "→E"] by blast
7386  AOT_hence ‹□∀y (∀F ([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7387    using "∀E" by blast
7388  AOT_hence ‹∀y □(∀F ([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7389    using CBF[THEN "→E"] by blast
7390  AOT_hence ‹□(∀F ([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7391    using "∀E" by blast
7392  AOT_hence ‹□∀F ([F]x ≡ [F]y) → □(φ{x} ≡ φ{y})›
7393    using "qml:1"[axiom_inst] "vdash-properties:6" by blast
7394  moreover AOT_assume ‹∀F([F]x ≡ [F]y)›
7395  ultimately AOT_show ‹□(φ{x} ≡ φ{y})› using "→E" "ind-nec" by blast
7396qed
7397
7398AOT_theorem "kirchner-thm-cor:2":
7399  ‹[λx1...xn φ{x1...xn}]↓ → ∀x1...∀xn∀y1...∀yn(∀F([F]x1...xn ≡ [F]y1...yn) → □(φ{x1...xn} ≡ φ{y1...yn}))›
7400proof(rule "→I"; rule GEN; rule GEN; rule "→I")
7401  fix x1xn y1yn
7402  AOT_assume ‹[λx1...xn φ{x1...xn}]↓›
7403  AOT_hence 0: ‹□∀x1...∀xn∀y1...∀yn (∀F ([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7404    by (rule "kirchner-thm:2"[THEN "≡E"(1)])
7405  AOT_have ‹∀x1...∀xn∀y1...∀yn □(∀F ([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7406  proof(rule GEN; rule GEN)
7407    fix x1xn y1yn
7408    AOT_show ‹□(∀F ([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7409      apply (rule "RM:1"[THEN "→E", rotated, OF 0]; rule "→I")
7410      using "∀E" by blast
7411  qed
7412  AOT_hence ‹∀y1...∀yn □(∀F ([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7413    using "∀E" by blast
7414  AOT_hence ‹□(∀F ([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7415    using "∀E" by blast
7416  AOT_hence ‹□(∀F ([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7417    using "∀E" by blast
7418  AOT_hence 0: ‹□∀F ([F]x1...xn ≡ [F]y1...yn) → □(φ{x1...xn} ≡ φ{y1...yn})›
7419    using "qml:1"[axiom_inst] "vdash-properties:6" by blast
7420  moreover AOT_assume ‹∀F([F]x1...xn ≡ [F]y1...yn)›
7421  moreover AOT_have ‹[λx1...xn □∀F ([F]x1...xn ≡ [F]y1...yn)]↓› by "cqt:2[lambda]"
7422  ultimately AOT_have ‹[λx1...xn □∀F ([F]x1...xn ≡ [F]y1...yn)]x1...xn ≡ [λx1...xn □∀F ([F]x1...xn ≡ [F]y1...yn)]y1...yn›
7423    using "∀E" by blast
7424  moreover AOT_have ‹[λx1...xn □∀F ([F]x1...xn ≡ [F]y1...yn)]y1...yn›
7425    apply (rule "β←C"(1))
7426      apply "cqt:2[lambda]"
7427     apply (fact "cqt:2[const_var]"[axiom_inst])
7428    by (simp add: RN GEN "oth-class-taut:3:a")
7429  ultimately AOT_have ‹[λx1...xn □∀F ([F]x1...xn ≡ [F]y1...yn)]x1...xn› using "≡E"(2) by blast
7430  AOT_hence ‹□∀F ([F]x1...xn ≡ [F]y1...yn)›
7431    using "β→C"(1) by blast
7432  AOT_thus ‹□(φ{x1...xn} ≡ φ{y1...yn})› using "→E" 0 by blast
7433qed
7434
7435AOT_define propositional :: ‹Π ⇒ φ› (‹Propositional'(_')›)
7436  "prop-prop1": ‹Propositional([F]) ≡df ∃p(F = [λy p])›
7437
7438AOT_theorem "prop-prop2:1": ‹∀p [λy p]↓›
7439  by (rule GEN) "cqt:2[lambda]"
7440
7441AOT_theorem "prop-prop2:2": ‹[λν φ]↓›
7442  by "cqt:2[lambda]"
7443
7444AOT_theorem "prop-prop2:3": ‹F = [λy p] → □∀x([F]x ≡ p)›
7445proof (rule "→I")
7446  AOT_assume 0: ‹F = [λy p]›
7447  AOT_show ‹□∀x([F]x ≡ p)›
7448    by (rule "rule=E"[rotated, OF 0[symmetric]]; rule RN; rule GEN; rule "beta-C-meta"[THEN "→E"])
7449      "cqt:2[lambda]"
7450qed
7451
7452AOT_theorem "prop-prop2:4": ‹Propositional([F]) → □Propositional([F])›
7453proof(rule "→I")
7454  AOT_assume ‹Propositional([F])›
7455  AOT_hence ‹∃p(F = [λy p])› using "≡dfE"[OF "prop-prop1"] by blast
7456  then AOT_obtain p where ‹F = [λy p]› using "∃E"[rotated] by blast
7457  AOT_hence ‹□(F = [λy p])› using "id-nec:2" "modus-tollens:1" "raa-cor:3" by blast
7458  AOT_hence ‹∃p □(F = [λy p])› using "∃I" by fast
7459  AOT_hence 0: ‹□∃p (F = [λy p])› by (metis Buridan "vdash-properties:10")
7460  AOT_show ‹□Propositional([F])›
7461    apply (AOT_subst ‹«Propositional([F])»› ‹«∃p (F = [λy p])»›)
7462     using "prop-prop1" "≡Df" apply presburger
7463    by (fact 0)
7464qed
7465
7466AOT_define indicriminate :: ‹Π ⇒ φ› ("Indiscriminate'(_')")
7467  "prop-indis": ‹Indiscriminate([F]) ≡df F↓ & □(∃x [F]x → ∀x [F]x)›
7468
7469AOT_theorem "prop-in-thm": ‹Propositional([Π]) → Indiscriminate([Π])›
7470proof(rule "→I")
7471  AOT_assume ‹Propositional([Π])›
7472  AOT_hence ‹∃p Π = [λy p]› using "≡dfE"[OF "prop-prop1"] by blast
7473  then AOT_obtain p where Π_def: ‹Π = [λy p]› using "∃E"[rotated] by blast
7474  AOT_show ‹Indiscriminate([Π])›
7475  proof (rule "≡dfI"[OF "prop-indis"]; rule "&I")
7476    AOT_show ‹Π↓›
7477      using Π_def by (meson "t=t-proper:1" "vdash-properties:6")
7478  next
7479    AOT_show ‹□(∃x [Π]x → ∀x [Π]x)›
7480    proof (rule "rule=E"[rotated, OF Π_def[symmetric]]; rule RN; rule "→I"; rule GEN)
7481      AOT_modally_strict {
7482        AOT_assume ‹∃x [λy p]x›
7483        then AOT_obtain a where ‹[λy p]a› using "∃E"[rotated] by blast
7484        AOT_hence 0: ‹p› by (metis "β→C"(1))
7485        AOT_show ‹[λy p]x› for x
7486          apply (rule "β←C"(1))
7487            apply "cqt:2[lambda]"
7488           apply (fact "cqt:2[const_var]"[axiom_inst])
7489          by (fact 0)
7490      }
7491    qed
7492  qed
7493qed
7494
7495AOT_theorem "prop-in-f:1": ‹Necessary([F]) → Indiscriminate([F])›
7496proof (rule "→I")
7497  AOT_assume ‹Necessary([F])›
7498  AOT_hence 0: ‹□∀x1...∀xn [F]x1...xn› using "≡dfE"[OF "contingent-properties:1"] by blast
7499  AOT_show ‹Indiscriminate([F])›
7500    by (rule "≡dfI"[OF "prop-indis"])
7501       (metis "0" "KBasic:1" "&I" "ex:1:a" "rule-ui:2[const_var]" "vdash-properties:6") 
7502qed
7503
7504AOT_theorem "prop-in-f:2": ‹Impossible([F]) → Indiscriminate([F])›
7505proof (rule "→I")
7506  AOT_modally_strict {
7507    AOT_have ‹∀x ¬[F]x → (∃x [F]x → ∀x [F]x)›
7508      by (metis "instantiation" "cqt-orig:3" "Hypothetical Syllogism" "deduction-theorem" "raa-cor:3")
7509  }
7510  AOT_hence 0: ‹□∀x ¬[F]x → □(∃x [F]x → ∀x [F]x)›
7511    by (rule "RM:1")
7512  AOT_assume ‹Impossible([F])›
7513  AOT_hence ‹□∀x ¬[F]x› using "≡dfE"[OF "contingent-properties:2"] "&E" by blast
7514  AOT_hence 1: ‹□(∃x [F]x → ∀x [F]x)› using 0 "→E" by blast
7515  AOT_show ‹Indiscriminate([F])›
7516    by (rule "≡dfI"[OF "prop-indis"]; rule "&I")
7517       (simp add: "ex:1:a" "rule-ui:2[const_var]" 1)+
7518qed
7519
7520AOT_theorem "prop-in-f:3:a": ‹¬Indiscriminate([E!])›
7521proof(rule "raa-cor:2")
7522  AOT_assume ‹Indiscriminate([E!])›
7523  AOT_hence 0: ‹□(∃x [E!]x → ∀x [E!]x)›
7524    using "≡dfE"[OF "prop-indis"] "&E" by blast
7525  AOT_hence ‹◇∃x [E!]x → ◇∀x [E!]x›
7526    using "KBasic:13" "vdash-properties:10" by blast
7527  moreover AOT_have ‹◇∃x [E!]x›
7528    by (simp add: "thm-cont-e:3")
7529  ultimately AOT_have ‹◇∀x [E!]x›
7530    by (metis "vdash-properties:6")
7531  AOT_thus ‹p & ¬p› for p
7532    by (metis "≡dfE" "conventions:5" "o-objects-exist:5" "reductio-aa:1")
7533qed
7534
7535AOT_theorem "prop-in-f:3:b": ‹¬Indiscriminate([E!]-)›
7536proof (rule "rule=E"[rotated, OF "rel-neg-T:2"[symmetric]]; rule "raa-cor:2")
7537  AOT_assume ‹Indiscriminate([λx ¬[E!]x])›
7538  AOT_hence 0: ‹□(∃x [λx ¬[E!]x]x → ∀x [λx ¬[E!]x]x)›
7539    using "≡dfE"[OF "prop-indis"] "&E" by blast
7540  AOT_hence ‹□∃x [λx ¬[E!]x]x → □∀x [λx ¬[E!]x]x›
7541    using "→E" "qml:1" "vdash-properties:1[2]" by blast
7542  moreover AOT_have ‹□∃x [λx ¬[E!]x]x›
7543    apply (AOT_subst ‹λκ. «[λx ¬[E!]x]κ»› ‹λκ. «¬[E!]κ»›)
7544    apply (rule "beta-C-meta"[THEN "→E"])
7545     apply "cqt:2[lambda]"
7546    by (metis (full_types) "B◇" RN "T◇" "cqt-further:2" "o-objects-exist:5" "vdash-properties:10")
7547  ultimately AOT_have 1: ‹□∀x [λx ¬[E!]x]x›
7548    by (metis "vdash-properties:6")
7549  AOT_have ‹□∀x ¬[E!]x›
7550    apply (AOT_subst_rev ‹λκ. «[λx ¬[E!]x]κ»› ‹λκ. «¬[E!]κ»›)
7551    apply (rule "beta-C-meta"[THEN "→E"])
7552     apply "cqt:2[lambda]"
7553    by (fact 1)
7554  AOT_hence ‹∀x □¬[E!]x› by (metis "CBF" "vdash-properties:10")
7555  moreover AOT_obtain a where abs_a: ‹O!a›
7556    using "instantiation" "o-objects-exist:1" "qml:2" "vdash-properties:1[2]" "vdash-properties:6" by blast
7557  ultimately AOT_have ‹□¬[E!]a› using "∀E" by blast
7558  AOT_hence 2: ‹¬◇[E!]a› by (metis "≡dfE" "conventions:5" "reductio-aa:1")
7559  AOT_have ‹A!a›
7560    apply (rule "=dfI"(2)[OF AOT_abstract])
7561     apply "cqt:2[lambda]"
7562    apply (rule "β←C"(1))
7563      apply "cqt:2[lambda]"
7564    using "cqt:2[const_var]"[axiom_inst] apply blast
7565    by (fact 2)
7566  AOT_thus ‹p & ¬p› for p using abs_a
7567    by (metis "≡E"(1) "oa-contingent:2" "reductio-aa:1")
7568qed
7569
7570AOT_theorem "prop-in-f:3:c": ‹¬Indiscriminate(O!)›
7571proof(rule "raa-cor:2")
7572  AOT_assume ‹Indiscriminate(O!)›
7573  AOT_hence 0: ‹□(∃x O!x → ∀x O!x)›
7574    using "≡dfE"[OF "prop-indis"] "&E" by blast
7575  AOT_hence ‹□∃x O!x → □∀x O!x›
7576    using "qml:1"[axiom_inst] "vdash-properties:6" by blast
7577  moreover AOT_have ‹□∃x O!x›
7578    using "o-objects-exist:1" by blast
7579  ultimately AOT_have ‹□∀x O!x›
7580    by (metis "vdash-properties:6")
7581  AOT_thus ‹p & ¬p› for p
7582    by (metis "o-objects-exist:3" "qml:2" "raa-cor:3" "vdash-properties:10" "vdash-properties:1[2]")
7583qed
7584
7585AOT_theorem "prop-in-f:3:d": ‹¬Indiscriminate(A!)›
7586proof(rule "raa-cor:2")
7587  AOT_assume ‹Indiscriminate(A!)›
7588  AOT_hence 0: ‹□(∃x A!x → ∀x A!x)›
7589    using "≡dfE"[OF "prop-indis"] "&E" by blast
7590  AOT_hence ‹□∃x A!x → □∀x A!x›
7591    using "qml:1"[axiom_inst] "vdash-properties:6" by blast
7592  moreover AOT_have ‹□∃x A!x›
7593    using "o-objects-exist:2" by blast
7594  ultimately AOT_have ‹□∀x A!x›
7595    by (metis "vdash-properties:6")
7596  AOT_thus ‹p & ¬p› for p
7597    by (metis "o-objects-exist:4" "qml:2" "raa-cor:3" "vdash-properties:10" "vdash-properties:1[2]")
7598qed
7599
7600AOT_theorem "prop-in-f:4:a": ‹¬Propositional(E!)›
7601  using "modus-tollens:1" "prop-in-f:3:a" "prop-in-thm" by blast
7602
7603AOT_theorem "prop-in-f:4:b": ‹¬Propositional(E!-)›
7604  using "modus-tollens:1" "prop-in-f:3:b" "prop-in-thm" by blast
7605
7606AOT_theorem "prop-in-f:4:c": ‹¬Propositional(O!)›
7607  using "modus-tollens:1" "prop-in-f:3:c" "prop-in-thm" by blast
7608
7609AOT_theorem "prop-in-f:4:d": ‹¬Propositional(A!)›
7610  using "modus-tollens:1" "prop-in-f:3:d" "prop-in-thm" by blast
7611
7612AOT_theorem "prop-prop-nec:1": ‹◇∃p (F = [λy p]) → ∃p(F = [λy p])›
7613proof(rule "→I")
7614  AOT_assume ‹◇∃p (F = [λy p])›
7615  AOT_hence ‹∃p ◇(F = [λy p])›
7616    by (metis "BF◇" "vdash-properties:10")
7617  then AOT_obtain p where ‹◇(F = [λy p])› using "∃E"[rotated] by blast
7618  AOT_hence ‹F = [λy p]› by (metis "derived-S5-rules:2" emptyE "id-nec:2" "vdash-properties:6")
7619  AOT_thus ‹∃p(F = [λy p])› by (rule "∃I")
7620qed
7621
7622AOT_theorem "prop-prop-nec:2": ‹∀p (F ≠ [λy p]) → □∀p(F ≠ [λy p])›
7623proof(rule "→I")
7624  AOT_assume ‹∀p (F ≠ [λy p])›
7625  AOT_hence ‹(F ≠ [λy p])› for p
7626    using "∀E" by blast
7627  AOT_hence ‹□(F ≠ [λy p])› for p
7628    by (rule "id-nec2:2"[unvarify β, THEN "→E", rotated]) "cqt:2[lambda]"
7629  AOT_hence ‹∀p □(F ≠ [λy p])› by (rule GEN)
7630  AOT_thus ‹□∀p (F ≠ [λy p])› using BF[THEN "→E"] by fast
7631qed
7632
7633AOT_theorem "prop-prop-nec:3": ‹∃p (F = [λy p]) → □∃p(F = [λy p])›
7634proof(rule "→I")
7635  AOT_assume ‹∃p (F = [λy p])›
7636  then AOT_obtain p where ‹(F = [λy p])› using "∃E"[rotated] by blast
7637  AOT_hence ‹□(F = [λy p])› by (metis "id-nec:2" "vdash-properties:6")
7638  AOT_hence ‹∃p□(F = [λy p])› by (rule "∃I")
7639  AOT_thus ‹□∃p(F = [λy p])› by (metis Buridan "vdash-properties:10")
7640qed
7641
7642AOT_theorem "prop-prop-nec:4": ‹◇∀p (F ≠ [λy p]) → ∀p(F ≠ [λy p])›
7643proof(rule "→I")
7644  AOT_assume ‹◇∀p (F ≠ [λy p])›
7645  AOT_hence ‹∀p ◇(F ≠ [λy p])› by (metis "Buridan◇" "vdash-properties:10")
7646  AOT_hence ‹◇(F ≠ [λy p])› for p
7647    using "∀E" by blast
7648  AOT_hence ‹F ≠ [λy p]› for p
7649    by (rule "id-nec2:3"[unvarify β, THEN "→E", rotated]) "cqt:2[lambda]"
7650  AOT_thus ‹∀p (F ≠ [λy p])› by (rule GEN)
7651qed
7652
7653AOT_theorem "enc-prop-nec:1": ‹◇∀F (x[F] → ∃p(F = [λy p])) → ∀F(x[F] → ∃p (F = [λy p]))›
7654proof(rule "→I"; rule GEN; rule "→I")
7655  fix F
7656  AOT_assume ‹◇∀F (x[F] → ∃p(F = [λy p]))›
7657  AOT_hence ‹∀F ◇(x[F] → ∃p(F = [λy p]))›
7658    using "Buridan◇" "vdash-properties:10" by blast
7659  AOT_hence 0: ‹◇(x[F] → ∃p(F = [λy p]))› using "∀E" by blast
7660  AOT_assume ‹x[F]›
7661  AOT_hence ‹□x[F]› by (metis "en-eq:2[1]" "≡E"(1))
7662  AOT_hence ‹◇∃p(F = [λy p])›
7663    using 0 by (metis "KBasic2:4" "≡E"(1) "vdash-properties:10")
7664  AOT_thus ‹∃p(F = [λy p])›
7665    using "prop-prop-nec:1"[THEN "→E"] by blast
7666qed
7667
7668AOT_theorem "enc-prop-nec:2": ‹∀F (x[F] → ∃p(F = [λy p])) → □∀F(x[F] → ∃p (F = [λy p]))›
7669  using "derived-S5-rules:1"[where Γ="{}", simplified, OF "enc-prop-nec:1"]
7670  by blast
7671
7672(*<*)
7673end
7674(*>*)